WEBVTT
Kind: captions
Language: en
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Alright. Sine and Cosine.
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Sine and cosine and tangent
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are all about the ratios
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and right triangles.
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You have a triangle and,
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since, right triangles
00:00:13.800 --> 00:00:15.800
right triangles can very easily be similar
00:00:15.800 --> 00:00:17.800
because they all have a right angle.
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You just need another angle
00:00:19.800 --> 00:00:21.800
and then triangles are similar.
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Imagine you have a right triangle
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with a bigger right triangle.
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Both have 90 degree angles,
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that's what makes them right triangles.
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And, now you need another angle.
00:00:36.200 --> 00:00:38.200
So, let's call that angle 30 degrees.
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And, the third angle has to automatically be equal,
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because the angles always have to add up to 180.
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So, once these two are equal
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and these two are equal, then
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those two have to be equal.
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So, in this case they both have to be 60.
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Then the idea of trig is
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the ratio of any side
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and you have a side here
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is the same as the ratio from here to here.
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on any size right triangle
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as long as they have that 30 degree angle.
00:01:06.700 --> 00:01:08.700
So, we just give names to the ratios.
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The names of the ratios are sine,
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and cosine, and tangent
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There is actually a reason why we call them sine, cosine and tangent.
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But, I will not go into why we call them sine, cosine and tangent.
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Okay. Um.
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So,
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let's remember how we define those.
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I am going to erase
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ah, I'll leave that there.
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So, we have a right triangle.
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We call this angle theta
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because that is a Greek letter. We can call it x.
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We can call it anything we want.
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Then the sine of that angle
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is the opposite side over the hypotenuse.
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Okay?
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Because the sine
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is the opposite over the hypotenuse.
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The cosine
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is the adjacent over the hypotenuse.
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Tangent
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is the opposite over the adjacent.
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That is how we have that mneumonic
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SOH CAH TOA.
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You all should have seen at some point in your life.
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You are seeing it again.
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Okay. By the way, you could go on in biology
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and never see trigonometry.
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I don't know how far you are going to go.
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You could go to med school and you are not going to find trigonometry.
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Um.
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Okay, so. This helps us
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name the ratio.
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There is also some problems
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lets step back a minutes.
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The three triangles
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Can you guys see over here?
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When I write on that board? Alright.
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So, the 30, 60, 90 triangle.
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In order to choose a 30, 60, 90 triangle
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so that each side are always the same proportion.
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So, the 30, 60, 90 triangle
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whatever side is opposite
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the 30 degrees is always
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half the hypotenuse.
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If this is 10 the hypotenuse is 20.
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If this is 100 the hypotenuse
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is 200. If this is 4
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the hypotenuse is 8. It is always half of
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what the hypotenuse is.
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The third side just comes out of the Pythagorean Theorem.
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It is the square root of 3.
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Okay?
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And that is true, so here, the proportion
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is the same.
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So, let's say this is 10.
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And, that would be 20, and that would be
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10 times the square root of 3.
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Now, that means that the sine is
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always going to have the same ratio.
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So, sine is 30 degrees is always going to come out a half.
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The cosine of 30 degrees
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Is always going to come square root of 3 over 2.
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It doesn't matter what the sides are,
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because the ratio will come out the same.
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So, whatever you put in,
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you simplify and reduce.
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It is always going to come out the same number.
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And, the tangent
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will always be 1 over the square root of 3.
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(Laughter)
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Alright. The sine of 60 degrees
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I was sitting with some professors yesterday
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and they were complaining about students.
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So, I will treat you as adults.
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Sine of 60 degrees
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square root of 3 over 2.
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Cosine of 60 degrees
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is a half.
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Like I said, these two are the
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opposite of each other. Okay? The sine of one
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is the cosine of the other
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Cosine of one is the sine of the other.
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Then, the tangent of 60 degrees
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Well, you go tangent of 60 degrees.
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And, you say, its square root of 3 over 1.
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Here if you do 10 square root of 3
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over 10, notice what happens...
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The 10s just cancel and you get the square root of 3 again.
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Okay? So, that ratio doesn't change.
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Now we will go to another type of triangle.
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These are called the special angles.
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These are the ones you are expected to memorize.
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Because, you don't know the sine
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of 73 degrees. I don't know the sine of
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73 degrees. You need a calculator for that.
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You don't need a calculator for these.
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These are the ones that we put on tests.
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Okay? If we had a
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73 degree angle on a test
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your answer would be something like sine of 73.
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And, I expect you to know that's
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you know 8 something.
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Okay? But if we say,
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sine of 60 degrees, you are supposed to know that it
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is the square root of 3 over 2.
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If you have
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A right triangle where these 2 sides are the same,
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the two legs,
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then the two angles are also the same.
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They are both going to be 45 degrees.
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And, the ratio of these two sides
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are only 1 because they are the same
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and that is the square root of 2.
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You can find sine and cosine and tangent of 45 degrees.
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The sine of 45 degrees and cosine of 45 degrees
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they are going to be the same.
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Alright. Because if you look at
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45 degree angle the opposite side
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and the adjacent side are the same.
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So, the sine and the cosine are going to come out the same.
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They are both going to be 1 over the square root of 2.
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For no special reason, we simplify this
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the square root of 2 over 2.
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I don't think that is more simple.
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In fact, if we had to square it. This is
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messier, and this is more simple. But,
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this fits nicely into a number line.
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Okay?
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And, the tangent of 45 degrees,
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On the tangent of 45
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is the opposite over the adjacent.
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They are both 1, so, that is just 1.
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So, then I came up with a nice handy way
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for everyone to memorize this.
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Okay. Sine and cosine are always over 2.
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So, this is 1,
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2
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3, and that goes
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3, 2, 1
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Did any of your teachers teach you this in school?
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One. That's it?
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That small a group?
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Okay. Because this used to be the way to learn it.
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And, then they came up with this unit circle stuff.
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And, that doesn't work as well.
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Because then you guys never memorize this.
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And, it isn't as hard
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if you get this stuff ingrained.
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Then you don't have to think about it. You know where it
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comes from and you get this stuff ingrained
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and it stays. Just like when
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you learn a language or a sport
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or music. Okay?
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So, the sine
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okay, so,
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tangent turns out
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to be sine over cosine
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I will show you why in a minute.
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Okay, so, that is your chart.
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Um.
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So, memorize this. As I said
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to you the other day, what you do when
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you walk into the exam
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pick out a corner of a back page
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or wherever you want to put it.
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So, that you dont have to think
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about the sine and cosine and tangent.
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Okay?
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Now let's just have a little more fun with this.
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Alright.
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Ah. Remember that the sine
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of that angle theta, A over C,
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opposite over hypotenuse.
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The cosine
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theta, B over C,
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and the tangent of theta, we wrote this already.
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The tangent of theta
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A over B.
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If you notice, if you take these two
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and divide them. You take A over C
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and you divide by B over C,
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you get
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A over B
00:10:03.300 --> 00:10:05.300
The sine of the angle
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divided by the cosine of the angle.
00:10:07.300 --> 00:10:09.300
And, that is going to be very useful. It is
00:10:09.300 --> 00:10:11.300
going to show up a lot. Okay.
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What happens is I will give you the sine, I will give you the cosine
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and I will say, find the tangent. Just divide them
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You know that A squared plus B squared
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in this triangle,
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equals C squared, right?
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Good. Well let's take everything and divide it by C squared.
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What is C squared divided by C squared?
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One.
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Well, you are in for another.
00:10:40.800 --> 00:10:42.800
So, neat, because we write this as
00:10:42.800 --> 00:10:44.800
A over C squared
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and B over C squared
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equals 1.
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Okay.
00:10:51.200 --> 00:10:53.200
So, A over C
00:10:53.200 --> 00:10:55.000
is the sine
00:10:55.000 --> 00:10:57.200
and B over C is the cosine.
00:10:57.200 --> 00:10:59.200
So, we can replace this
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with sine of theta squared
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plus cosine of theta
00:11:03.200 --> 00:11:05.200
squared
00:11:05.200 --> 00:11:07.200
equals 1.
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That's the second thing to memorize.
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Over on this board.
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Sine squared of any angle
00:11:15.500 --> 00:11:18.200
plus cosine squared of the same angle
00:11:18.200 --> 00:11:20.200
will be 1.
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That is the Pythagorean Theorem
00:11:22.200 --> 00:11:24.200
in the trigonometry world.
00:11:24.200 --> 00:11:27.000
Okay? We write the squared here
00:11:27.000 --> 00:11:30.000
because although it is sine theta squared,
00:11:32.000 --> 00:11:34.000
like this,
00:11:34.000 --> 00:11:38.000
then it is not clear if I am squaring theta or squaring sine.
00:11:38.000 --> 00:11:40.500
By putting it here,
00:11:40.500 --> 00:11:42.500
that tells you I am
00:11:42.500 --> 00:11:44.500
squaring sine, not the
00:11:44.500 --> 00:11:46.500
angle. If you do it
00:11:46.500 --> 00:11:48.500
on your calculator, you do sine of
00:11:48.500 --> 00:11:50.500
the angle closed parentheses
00:11:50.500 --> 00:11:52.500
then you square it.
00:11:52.500 --> 00:11:54.500
If you square it before you
00:11:54.500 --> 00:11:56.500
close the parentheses it will square the angle.
00:11:56.500 --> 00:11:58.500
Okay? When you use the
00:11:58.500 --> 00:12:00.500
calculator and you want
00:12:00.500 --> 00:12:02.500
to do sine of an angle, close
00:12:02.500 --> 00:12:04.500
parentheses, then square it.
00:12:04.500 --> 00:12:06.500
Also, you using your calculator
00:12:06.500 --> 00:12:08.500
make sure you are in degree mode.
00:12:09.000 --> 00:12:11.000
The calculators have more than one mode.
00:12:11.000 --> 00:12:13.000
And often if you are having
00:12:13.000 --> 00:12:15.000
problems you are in the wrong mode.
00:12:15.000 --> 00:12:17.000
Okay?
00:12:17.000 --> 00:12:19.000
So far, so good?
00:12:21.000 --> 00:12:23.000
Okay.
00:12:23.000 --> 00:12:25.000
I was telling you guys the other day.
00:12:25.000 --> 00:12:27.000
You are ancient Babylonians
00:12:27.000 --> 00:12:29.000
and you want to knock down a wall
00:12:29.000 --> 00:12:31.000
you have to figure out how high it is.
00:12:31.000 --> 00:12:33.000
So, a person stands 40 feet from the base of a wall.
00:12:33.000 --> 00:12:35.000
That wasn't far enough back then, though, right?
00:12:35.000 --> 00:12:37.000
It would be a lot farther.
00:12:39.000 --> 00:12:41.000
40 feet from the base of the wall
00:12:41.000 --> 00:12:43.000
and you measure the angle of elevation,
00:12:43.000 --> 00:12:45.000
this angle, and it is 60 degrees.
00:12:45.000 --> 00:12:47.000
How tall is the wall?
00:12:49.000 --> 00:12:51.000
So, let's set it up.
00:12:55.000 --> 00:12:57.000
Okay, this,
00:12:57.000 --> 00:12:59.000
we are standing to face the wall
00:12:59.000 --> 00:13:01.000
we are 40 feet away.
00:13:01.000 --> 00:13:03.000
We measure that as 60 degrees.
00:13:05.000 --> 00:13:07.000
Right. So, how tall is this?
00:13:09.000 --> 00:13:11.000
So, let's use a trig function.
00:13:11.000 --> 00:13:13.000
So, which trig function would we use?
00:13:17.000 --> 00:13:19.000
Tangent.
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Good. So, we have the adjacent
00:13:21.000 --> 00:13:23.000
side and we want to find
00:13:23.000 --> 00:13:25.000
the opposite side. Tangent
00:13:25.000 --> 00:13:27.000
is opposite over adjacent.
00:13:27.000 --> 00:13:29.500
You look at this and say, "Well, tangent of 60 degrees
00:13:31.500 --> 00:13:36.000
is the opposite over the adjacent.
00:13:40.000 --> 00:13:42.000
Cross multiply.
00:13:43.000 --> 00:13:45.000
And you get 40
00:13:46.000 --> 00:13:48.000
times the tangent of 60
00:13:49.000 --> 00:13:51.000
is x.
00:13:51.500 --> 00:13:54.000
Now, what is the tangent of 60?
00:13:56.500 --> 00:13:58.500
Square root of 3.
00:13:59.500 --> 00:14:01.000
So, 40
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times the square root of 3
00:14:04.000 --> 00:14:06.500
is x. Okay? Yes.
00:14:08.000 --> 00:14:10.500
Why do I use tangent? Alright.
00:14:10.500 --> 00:14:12.500
If I am looking at the wall, and I have this distance,
00:14:12.500 --> 00:14:14.500
that is the 40 feet from the bottom
00:14:14.500 --> 00:14:17.000
of the wall. And I want to know how high the wall is.
00:14:17.000 --> 00:14:19.000
So, I have my adjacent side
00:14:19.000 --> 00:14:21.000
that is the one that is touching my feet.
00:14:21.000 --> 00:14:23.000
Student: How about the other side?
00:14:23.000 --> 00:14:25.000
I don't know what my hypotenuse is and I don't care what my
00:14:25.000 --> 00:14:27.000
hypotenuse is. So, which
00:14:27.000 --> 00:14:29.000
trig function uses opposite and
00:14:29.000 --> 00:14:31.000
adjacent? Tangent.
00:14:31.000 --> 00:14:33.000
Okay?
00:14:33.000 --> 00:14:35.000
Any other questions?
00:14:35.000 --> 00:14:37.000
Feel free to ask.
00:14:39.000 --> 00:14:41.000
Yes.
00:14:41.800 --> 00:14:43.800
Okay. So let us do that again. (laughter)
00:14:43.800 --> 00:14:45.800
That is alright. This is the hardest part
00:14:45.800 --> 00:14:47.800
of trigonometry. How do you
00:14:47.800 --> 00:14:49.800
know which trig function to use.
00:14:49.800 --> 00:14:51.800
First question to ask is:
00:14:51.800 --> 00:14:53.800
do you want to know the hypotenuse?
00:14:53.800 --> 00:14:55.800
Do you want to know the angle distance?
00:14:55.800 --> 00:14:58.500
Or a side distance?
00:14:58.500 --> 00:15:00.500
If I don't knowthe hypotenuse,
00:15:00.500 --> 00:15:02.500
I have to use tangent.
00:15:02.500 --> 00:15:04.500
Because there are only 3 things you know.
00:15:08.500 --> 00:15:10.500
SOH CAH TOA
00:15:12.500 --> 00:15:14.500
Sine and cosine both use the
00:15:14.500 --> 00:15:16.500
hypotenuse. If I don't know the hypotenuse,
00:15:16.500 --> 00:15:18.500
I have to use tangent. Okay?
00:15:18.500 --> 00:15:20.500
So, opposite side is
00:15:20.500 --> 00:15:22.500
I am looking at the triangle, standing at the
00:15:22.500 --> 00:15:24.500
corner and it is
00:15:24.500 --> 00:15:26.500
the one that is directly opposite me.
00:15:26.500 --> 00:15:29.000
The adjacent side is the one under my feet.
00:15:29.000 --> 00:15:31.000
The tangent is
00:15:31.000 --> 00:15:33.000
the one shooting the bow and arrow
00:15:33.000 --> 00:15:35.000
the tangent would be the diagonal.
00:15:35.000 --> 00:15:37.000
Where the arrow goes. If I have,
00:15:37.000 --> 00:15:39.000
a rope I am
00:15:39.000 --> 00:15:41.000
holding up to a flag pole
00:15:41.000 --> 00:15:43.000
or something. That would be the diagonal.
00:15:43.000 --> 00:15:45.000
Okay. That is the hypotenuse.
00:15:45.000 --> 00:15:47.000
So, if you don't know the hypotenuse, I
00:15:47.000 --> 00:15:49.000
have to use tangent. Okay?
00:15:50.200 --> 00:15:52.000
We will practice this again.
00:15:53.000 --> 00:15:56.000
So, one side of a triangle, I know that I am looking at
00:15:56.000 --> 00:15:58.000
the side opposite me. And, I have the side
00:15:58.000 --> 00:16:00.000
adjacent to me. That is why
00:16:00.000 --> 00:16:02.000
I am using tangent. Okay,
00:16:02.000 --> 00:16:04.000
so, I say, "Well, what is the tangent?" The tangent is
00:16:04.000 --> 00:16:06.000
that opposite side divided by
00:16:06.000 --> 00:16:08.000
this adjacent side. That x
00:16:08.000 --> 00:16:10.000
divided by the 40.
00:16:10.000 --> 00:16:12.300
And, I know that is equal to the square root of 3,
00:16:12.300 --> 00:16:14.000
because the tangent of 60
00:16:14.000 --> 00:16:16.000
is the square root of 3.
00:16:16.000 --> 00:16:18.000
From the chart.
00:16:18.000 --> 00:16:20.000
Which you are going to, the chart,
00:16:20.000 --> 00:16:22.000
which you are going to write down on your test
00:16:22.000 --> 00:16:24.000
so you don't even have to think about it.
00:16:26.000 --> 00:16:29.000
Okay. Sure. You want me to do it one more time?
00:16:29.500 --> 00:16:32.000
Yes. One more time?
00:16:32.000 --> 00:16:34.000
Okay.
00:16:34.000 --> 00:16:36.000
Um.
00:16:36.000 --> 00:16:38.000
So, once more. When you are
00:16:38.000 --> 00:16:40.000
given the word problem you have to
00:16:40.000 --> 00:16:42.500
think about what we give you and what we don't give you.
00:16:42.500 --> 00:16:44.500
Okay? So, I will give you
00:16:44.500 --> 00:16:46.200
another word problem. In a second
00:16:46.200 --> 00:16:48.000
and you will see the difference.
00:16:48.000 --> 00:16:50.000
So, I give you how
00:16:50.000 --> 00:16:52.000
far I am away from a wall and where I am standing.
00:16:52.000 --> 00:16:54.000
See I am not standing at the top of the wall and looking down.
00:16:54.000 --> 00:16:56.500
I am below the wall and looking up.
00:16:56.500 --> 00:17:00.000
The adjacent side is the distance from below my feet
00:17:00.000 --> 00:17:02.000
to the bottom of the wall.
00:17:02.000 --> 00:17:04.000
Everyone understand that? Adjacent means next to.
00:17:04.000 --> 00:17:06.000
Yes.
00:17:10.500 --> 00:17:13.500
Not quite the angle of the direction.
00:17:13.500 --> 00:17:15.500
Okay, so, if you
00:17:15.500 --> 00:17:17.500
hold your hand out straight and you are looking at something.
00:17:17.500 --> 00:17:20.000
And, you elevate your head as the angle of elevation.
00:17:20.000 --> 00:17:22.000
You drop your head as the angle of depression.
00:17:22.000 --> 00:17:24.000
You are depressing your head. So,
00:17:25.300 --> 00:17:28.000
that would be the angle of depression. So, the angle of depression
00:17:28.000 --> 00:17:30.000
is almost the same thing as the angle of elevation.
00:17:30.000 --> 00:17:31.500
Okay?
00:17:31.500 --> 00:17:33.500
So, there is 2 ways to measure things.
00:17:33.500 --> 00:17:35.500
So, if you are in an airplane and you want to land.
00:17:35.500 --> 00:17:37.500
Right. So, you are flying straight and now you are going to
00:17:37.500 --> 00:17:39.500
dip down, so, now you are depressing the angle.
00:17:39.500 --> 00:17:41.500
That is the angle of depression.
00:17:41.500 --> 00:17:43.500
Okay?
00:18:09.500 --> 00:18:11.500
Um,
00:18:11.500 --> 00:18:13.500
so, again.
00:18:13.500 --> 00:18:15.500
So, you look at where you are standing and you imagine
00:18:15.500 --> 00:18:17.500
a triangle and you say, "I am standing here."
00:18:17.500 --> 00:18:20.000
I will draw a really good representation of me.
00:18:21.000 --> 00:18:23.000
My own work.
00:18:23.000 --> 00:18:25.000
And I say, "My distance from
00:18:25.000 --> 00:18:27.000
the bottom of the wall. Well, that is
00:18:27.000 --> 00:18:29.000
adjacent from where I am. That is the
00:18:29.000 --> 00:18:31.000
adjacent side. This side is
00:18:31.000 --> 00:18:33.000
opposite me. Because,
00:18:33.000 --> 00:18:35.000
it is opposite me. This is
00:18:35.000 --> 00:18:37.000
the hypotenuse. I have no idea what the
00:18:37.000 --> 00:18:39.000
hypotenuse is." Okay?
00:18:39.000 --> 00:18:41.000
So, now that I know
00:18:41.000 --> 00:18:43.000
the adjacent, and I am looking for the opposite,
00:18:43.000 --> 00:18:45.000
I am going to use tangent.
00:18:45.000 --> 00:18:47.000
Okay?
00:18:47.000 --> 00:18:49.000
Then, I set it up. I say, "Well, the
00:18:49.000 --> 00:18:51.000
tangent of that angle is the side
00:18:51.000 --> 00:18:53.000
opposite divided by the distance
00:18:53.000 --> 00:18:55.000
of the adjacent side.
00:18:55.000 --> 00:18:57.500
I cross multiply. Because, we love cross multiplying.
00:18:57.500 --> 00:18:59.500
By the way, you can cross multiply
00:18:59.500 --> 00:19:01.500
in one direction without going in the other direction.
00:19:01.500 --> 00:19:03.500
Then you don't have to do both.
00:19:03.500 --> 00:19:06.000
But, if you wanted to you can imagine there is a 1 down there.
00:19:07.000 --> 00:19:09.000
Now, that I cross multiplied, I know that
00:19:09.000 --> 00:19:11.500
x is 40 times the tangent of 60.
00:19:11.500 --> 00:19:13.800
The tangent of 60 is just the square root of 3.
00:19:16.200 --> 00:19:18.500
Okay. Let us think about another one.
00:19:26.500 --> 00:19:28.500
Okay,
00:19:28.500 --> 00:19:30.500
a pole
00:19:30.500 --> 00:19:33.000
you know that thing that sticks up straight.
00:19:33.000 --> 00:19:35.000
Is supported by a rope
00:19:35.000 --> 00:19:37.000
that runs from the top of the pole
00:19:37.000 --> 00:19:39.000
to the ground. The rope
00:19:39.000 --> 00:19:41.000
is 50 feet long, and
00:19:41.000 --> 00:19:43.000
I'm sorry, "fitty" feet long, (laughter)
00:19:45.000 --> 00:19:47.000
makes an angle with the ground of 30 degrees.
00:19:51.000 --> 00:19:53.000
How tall is the pole?
00:19:55.000 --> 00:19:57.000
Pole supported by a rope.
00:19:57.000 --> 00:19:59.000
well, from the top.
00:20:05.000 --> 00:20:07.000
Assuming the pole is standing straight up.
00:20:07.000 --> 00:20:08.200
i could leave that out.
00:20:08.200 --> 00:20:10.200
The pole could be tilted.
00:20:10.200 --> 00:20:12.200
But, assuming that the pole was standing straight up.
00:20:13.000 --> 00:20:15.200
And, rope runs from the top of the pole
00:20:15.200 --> 00:20:17.200
down to the ground. You are
00:20:17.200 --> 00:20:19.200
getting the hypotenuse. Okay?
00:20:19.200 --> 00:20:21.200
And, that rope is
00:20:22.200 --> 00:20:24.900
"Fitty" feet long. (laughter)
00:20:26.000 --> 00:20:28.000
Okay?
00:20:28.000 --> 00:20:30.000
And, the angle that it makes with the ground
00:20:30.000 --> 00:20:32.000
is 30 degrees.
00:20:32.000 --> 00:20:34.000
So, I want to know.
00:20:35.800 --> 00:20:38.000
How tall is the pole?
00:20:40.000 --> 00:20:42.000
Okay?
00:20:42.000 --> 00:20:44.000
Are we actually getting answers?
00:20:44.500 --> 00:20:46.000
Alrighty.
00:20:46.000 --> 00:20:48.000
Um. Well, you are correct,
00:20:48.000 --> 00:20:50.000
but let's see.
00:20:50.000 --> 00:20:52.000
What trig function do I use?
00:20:52.000 --> 00:20:53.500
Sine!
00:20:53.500 --> 00:20:55.500
How do I know I use sine? Because
00:20:55.500 --> 00:20:57.500
opposite over hypotenuse.
00:20:57.500 --> 00:20:59.500
This is great!
00:20:59.500 --> 00:21:01.500
Okay. I have the hypotenuse
00:21:01.500 --> 00:21:04.000
that is the rope. That is the diagonal length.
00:21:04.000 --> 00:21:06.000
I am looking for the opposite length.
00:21:06.000 --> 00:21:08.000
It will be sine
00:21:08.000 --> 00:21:10.000
of 30 degrees
00:21:10.000 --> 00:21:12.000
is x over 50.
00:21:14.000 --> 00:21:16.000
Then, I cross multiply,
00:21:16.000 --> 00:21:18.000
and I get 50
00:21:18.000 --> 00:21:20.000
times sine of 30
00:21:22.000 --> 00:21:24.000
equals x.
00:21:24.000 --> 00:21:26.000
Sine of 30 is a half.
00:21:26.000 --> 00:21:28.000
So, 50
00:21:28.000 --> 00:21:30.000
times a half.
00:21:30.000 --> 00:21:32.000
And, x is 25.
00:21:32.000 --> 00:21:34.000
Can everyone read that down there?
00:21:34.000 --> 00:21:36.000
That is not hidden, but it is low.
00:21:40.000 --> 00:21:42.000
Good. Alright.
00:21:43.000 --> 00:21:45.000
So, that's basic premise of this course.
00:21:45.000 --> 00:21:46.500
How do we make this harder?
00:21:46.500 --> 00:21:48.500
We don't use 30 degrees.
00:21:48.500 --> 00:21:50.000
We use something like 12 degrees.
00:21:50.000 --> 00:21:52.500
Then, you need a calculator. The principal doesn't change.
00:21:53.000 --> 00:21:55.000
So, in fact, because this is a special
00:21:55.000 --> 00:21:57.500
triangle we do not really need to use trigonometry.
00:21:57.500 --> 00:22:01.000
A lot of you just know about a 30 60 90 triangle.
00:22:01.000 --> 00:22:03.000
00:22:03.000 --> 00:22:05.000
But, you should get the principal.
00:22:05.000 --> 00:22:07.000
Because, as I say, what would you do if I gave you 12 degrees?
00:22:07.000 --> 00:22:09.000
Well, then the answer would be
00:22:09.000 --> 00:22:11.000
50 times the sin of 12.
00:22:11.000 --> 00:22:13.000
These would be alone.
00:22:13.000 --> 00:22:15.000
If I were to tell you it was an x equation.
00:22:15.000 --> 00:22:17.000
It would be 50 sine of x.
00:22:17.000 --> 00:22:19.000
okay?
00:22:19.000 --> 00:22:21.000
Got that? Should I repeat all of this?
00:22:22.000 --> 00:22:24.000
Another round. Okay.
00:22:24.000 --> 00:22:26.000
Once again. I am at the top of the pole.
00:22:26.000 --> 00:22:28.500
I have a rope that runs to the bottom of the pole.
00:22:28.500 --> 00:22:31.000
At the top of the pole, the rope runs down at an angle.
00:22:31.000 --> 00:22:34.000
So, that is going to be the hypotenuse of the triangle.
00:22:34.000 --> 00:22:36.000
Then, I tell you that the angle that the rope
00:22:36.000 --> 00:22:38.000
makes with the ground is 30 degrees.
00:22:38.000 --> 00:22:40.000
Okay?
00:22:40.000 --> 00:22:42.000
So, now that I know this distance.
00:22:42.000 --> 00:22:44.000
I am at this angle.
00:22:44.000 --> 00:22:47.000
I am looking for tall the pole is.
00:22:47.000 --> 00:22:49.500
That is x. The height of the pole.
00:22:49.500 --> 00:22:51.500
I can use sine.
00:22:52.000 --> 00:22:55.000
The sine uses opposite over hypotenuse.
00:22:55.000 --> 00:22:57.000
I wrote that somewhere. Ah, opposite over hypotenuse.
00:22:57.000 --> 00:22:58.500
Okay?
00:22:58.500 --> 00:23:00.500
So, I say that the sine of 30 degrees
00:23:00.500 --> 00:23:02.500
is the height of the pole
00:23:02.500 --> 00:23:04.500
divided by 50.
00:23:04.500 --> 00:23:06.500
I cross multiply so that
00:23:06.500 --> 00:23:08.500
50 times the sine of 30 degrees equals x.
00:23:09.500 --> 00:23:12.500
Then I just solve. Sine of 30 is a half.
00:23:12.500 --> 00:23:14.000
I know that there.
00:23:14.000 --> 00:23:16.000
50 times a half is 25.
00:23:17.500 --> 00:23:19.500
Okay.
00:23:19.500 --> 00:23:22.500
Alright. Another word problem or we will move on to more exciting stuff.
00:23:22.500 --> 00:23:24.500
Who wants another word problem?
00:23:24.500 --> 00:23:26.500
So, we are sitting here and we are saying, "Well, I know
00:23:26.500 --> 00:23:28.500
how to do the sine of 30 and 45
00:23:28.500 --> 00:23:30.500
and 60 and 12
00:23:30.500 --> 00:23:32.500
You can do anything in your right mind.
00:23:32.500 --> 00:23:34.500
Then, you say to yourself,
00:23:34.500 --> 00:23:36.500
what if I want to find the sine, the cosine, and tangent.
00:23:36.500 --> 00:23:39.000
Of something that is not in a right triangle.
00:23:39.000 --> 00:23:42.000
How do I...okay...so the angles
00:23:42.000 --> 00:23:45.000
in a right triangle are limited to being
00:23:45.000 --> 00:23:47.000
between 0 and 90 degrees.
00:23:47.000 --> 00:23:49.000
Why is that true?
00:23:49.000 --> 00:23:51.000
Think about a right triangle.
00:23:56.000 --> 00:23:59.000
This angle is 90 degrees.
00:24:00.000 --> 00:24:02.500
The other two angles have to add up to 90 degrees.
00:24:02.500 --> 00:24:05.000
Because the total for triangles is 180.
00:24:06.000 --> 00:24:07.500
So...
00:24:07.500 --> 00:24:09.500
these have to add up to 90.
00:24:09.500 --> 00:24:11.500
If 1 is 1 then the other is 89.
00:24:11.500 --> 00:24:13.700
If this is .1 than the other is 89.9.
00:24:13.700 --> 00:24:16.200
You can't get bigger than 90 degrees.
00:24:17.000 --> 00:24:19.500
So this is limited to only being able to find
00:24:19.500 --> 00:24:21.500
sines, cosines, and tangents
00:24:21.500 --> 00:24:24.000
of angles between 0 and 90.
00:24:29.000 --> 00:24:31.000
So, some very clever people said,
00:24:31.000 --> 00:24:33.000
"Well, let's see if we can find them of an angle
00:24:33.000 --> 00:24:35.000
more than 0."
00:24:35.000 --> 00:24:37.500
More than 90, sorry. or less than 0.
00:24:40.500 --> 00:24:42.500
What they did was they went off
00:24:42.500 --> 00:24:44.500
the coordinate axes and they said
00:24:44.500 --> 00:24:46.500
ok, let's pretend the angle
00:24:46.500 --> 00:24:48.500
I showed you this last time. But, we will do it again.
00:24:48.500 --> 00:24:51.200
The angle is say here.
00:24:51.200 --> 00:24:53.500
And, I want to find
00:24:53.500 --> 00:24:55.500
the sine, cosine, and tangent of that angle.
00:24:55.500 --> 00:24:58.000
But, that angle is more than 90 degrees.
00:24:58.000 --> 00:25:00.000
It is a hundred and something.
00:25:02.000 --> 00:25:04.000
We are looking for
00:25:04.000 --> 00:25:06.000
sine and cosine of this angle
00:25:06.000 --> 00:25:08.000
we will call theta.
00:25:10.000 --> 00:25:12.000
People looked around and said, "Well,
00:25:12.000 --> 00:25:14.000
I could just
00:25:16.000 --> 00:25:18.000
use this angle instead.
00:25:18.000 --> 00:25:20.000
180 minus theta.
00:25:24.000 --> 00:25:26.000
The opposite and adjacent
00:25:26.000 --> 00:25:28.000
do not make any sense.
00:25:28.000 --> 00:25:30.500
Why don't they make any sense?
00:25:30.500 --> 00:25:34.000
Well, because I do not have a 90 degree angle to work with.
00:25:34.500 --> 00:25:36.500
If you did this instead,
00:25:36.500 --> 00:25:38.500
let's just define it,
00:25:42.500 --> 00:25:44.500
as the coordinates of the circle.
00:25:44.500 --> 00:25:46.500
Oh, could you do that again.
00:25:46.500 --> 00:25:48.500
(laughter)
00:25:56.500 --> 00:26:02.000
So, why don't we just call sine and cosine the coordinates of that point.
00:26:02.000 --> 00:26:03.500
Why can we do that?
00:26:03.500 --> 00:26:05.500
I will do a little erasing.
00:26:05.500 --> 00:26:07.500
Well, we got the circle.
00:26:09.500 --> 00:26:12.000
Let us just look at the first quadrant for a minute.
00:26:14.500 --> 00:26:19.000
I got a triangle that is just touching.
00:26:24.000 --> 00:26:27.000
If I make the radius of this circle 1,
00:26:31.500 --> 00:26:35.000
the coordinates of this point are (x,y).
00:26:38.500 --> 00:26:41.000
And, now I have found some angle, theta.
00:26:41.000 --> 00:26:43.000
And, sine
00:26:43.000 --> 00:26:45.000
theta
00:26:45.000 --> 00:26:47.000
is what? Is the opposite over the hypotenuse.
00:26:47.000 --> 00:26:49.000
It is y divided by 1.
00:26:51.000 --> 00:26:53.000
I don't need to write the divided by 1. It is just 1.
00:26:53.000 --> 00:26:55.000
And, similarly
00:26:55.000 --> 00:26:57.000
if I want to find the
00:26:57.000 --> 00:26:59.000
x value, the cosine of theta
00:26:59.000 --> 00:27:01.000
is x divided by 1.
00:27:03.500 --> 00:27:06.000
So, the coordinates of that point
00:27:06.000 --> 00:27:09.500
are really (cos (theta), sin (theta)).
00:27:11.500 --> 00:27:13.500
00:27:13.500 --> 00:27:15.500
00:27:15.500 --> 00:27:17.500
00:27:17.500 --> 00:27:19.500
00:27:19.500 --> 00:27:21.500
So, any point on the unit circle
00:27:21.500 --> 00:27:25.700
the coordinates of the point are cosine of theta, sine of theta.
00:27:25.700 --> 00:27:28.500
This is very easy because now you can say
00:27:28.500 --> 00:27:30.800
okay now I can find sine and cosine of angles
00:27:30.800 --> 00:27:32.800
other than
00:27:32.800 --> 00:27:34.800
between 0 and 90 degrees.
00:27:34.800 --> 00:27:36.800
Now, let us go back to our example.
00:27:36.800 --> 00:27:38.800
I will draw another circle.
00:27:42.800 --> 00:27:44.800
Now, I find the sine
00:27:44.800 --> 00:27:46.500
and the cosine
00:27:46.500 --> 00:27:48.200
of this angle.
00:27:48.200 --> 00:27:50.800
But remember it is just the coordinates of this point.
00:27:57.200 --> 00:27:59.200
So, the coordinates of that point
00:27:59.200 --> 00:28:01.200
are this distance and this distance.
00:28:01.200 --> 00:28:04.200
It is the x distance and the y distance.
00:28:06.200 --> 00:28:08.200
If I wanted to find
00:28:08.200 --> 00:28:10.200
some angle down here.
00:28:17.000 --> 00:28:19.500
Again, they are just the cosine and sine at that point.
00:28:19.500 --> 00:28:22.000
For that angle.
00:28:22.000 --> 00:28:24.000
So, in general, if I wanted to find
00:28:24.000 --> 00:28:26.000
sine and cosine of angles
00:28:26.000 --> 00:28:28.000
other than between
00:28:28.000 --> 00:28:29.500
0 and 90 degrees
00:28:29.500 --> 00:28:31.500
I would put them on the unit circle.
00:28:31.500 --> 00:28:33.500
So, let us do an example.
00:28:39.500 --> 00:28:41.500
Let's find the sine
00:28:41.500 --> 00:28:43.500
of 120 degrees.
00:28:45.500 --> 00:28:47.500
It is better to have examples.
00:28:57.500 --> 00:28:59.500
Yeah, that is terrible.
00:29:07.500 --> 00:29:09.500
So where is 120 degrees.
00:29:09.500 --> 00:29:11.500
Well, I can start wherever I want.
00:29:11.500 --> 00:29:13.500
But it is tradition to say,
00:29:13.500 --> 00:29:17.000
0 degrees is on the positive x axis.
00:29:17.000 --> 00:29:19.000
That makes this 90,
00:29:19.000 --> 00:29:21.000
that makes 180
00:29:21.000 --> 00:29:23.000
and, that makes this.
00:29:24.200 --> 00:29:26.200
270 degrees, and then
00:29:26.200 --> 00:29:28.200
360 degrees
00:29:28.200 --> 00:29:30.200
is going all the way around the circle.
00:29:42.200 --> 00:29:44.200
So, where would 120 degrees be?
00:29:44.200 --> 00:29:46.200
Well, if this is 0
00:29:46.200 --> 00:29:48.200
and this is 90
00:29:48.200 --> 00:29:50.200
you just keep going until you get to about
00:29:50.200 --> 00:29:52.200
there.
00:29:56.400 --> 00:29:59.000
Okay? Imagine you are 0.
00:29:59.000 --> 00:30:00.500
And you rotate to 90.
00:30:00.500 --> 00:30:02.500
and you keep rotating until you get to 120.
00:30:04.000 --> 00:30:06.500
So, if I want to find the sine of 120 degrees.
00:30:06.500 --> 00:30:08.500
I need to the the sine and cosine
00:30:09.500 --> 00:30:11.500
This angle, well,
00:30:11.500 --> 00:30:13.500
that is going to be our theta.
00:30:15.200 --> 00:30:17.200
How big is theta?
00:30:17.200 --> 00:30:19.200
60 degrees. Why is it 60?
00:30:19.200 --> 00:30:21.200
Because that is 180 and that is 120.
00:30:21.200 --> 00:30:23.200
There is 60 degrees left over.
00:30:29.200 --> 00:30:31.200
So, what is the sine of 60 degrees?
00:30:35.200 --> 00:30:37.200
Well, it is radical 3 over 2.
00:30:39.200 --> 00:30:41.200
You have to be careful now.
00:30:41.200 --> 00:30:43.700
Somewhere over here in the second quadrant.
00:30:43.900 --> 00:30:45.900
So, this distance
00:30:45.900 --> 00:30:47.900
sine is still going to be positive.
00:30:47.900 --> 00:30:49.900
because it is still
00:30:49.900 --> 00:30:51.900
pointing up. The y direction
00:30:51.900 --> 00:30:54.500
is pointing up is positive and negative is down.
00:30:55.000 --> 00:30:57.000
If I wanted to find the
00:30:57.000 --> 00:30:59.000
x value the cosine
00:30:59.000 --> 00:31:01.000
is going to be negative because I am
00:31:01.000 --> 00:31:03.000
pointing this way on the x-axis.
00:31:03.300 --> 00:31:07.000
If I said, the cosine of 120 degrees,
00:31:07.000 --> 00:31:09.000
that would be the same as the
00:31:09.000 --> 00:31:11.000
cosine of 60 degrees.
00:31:11.000 --> 00:31:13.500
so it is negative.
00:31:13.500 --> 00:31:16.000
Cosine of 60 degrees is a half.
00:31:16.000 --> 00:31:18.000
So, this would be negative one half.
00:31:20.000 --> 00:31:22.000
Should I repeat that?
00:31:22.000 --> 00:31:24.000
Okay. You think about the coordinates
00:31:24.000 --> 00:31:26.000
on the x axis and the y axis.
00:31:26.000 --> 00:31:28.000
As long as I am pointing up,
00:31:28.000 --> 00:31:30.000
I have a positive value.
00:31:30.000 --> 00:31:32.000
Positive y value.
00:31:32.000 --> 00:31:34.000
So, the sine
00:31:34.000 --> 00:31:36.000
is the opposite of this angle.
00:31:37.000 --> 00:31:40.000
Sine is that length and that is a
00:31:40.000 --> 00:31:43.000
positive distance which is just up the y-axis.
00:31:43.000 --> 00:31:45.000
However, when I am going
00:31:45.000 --> 00:31:47.000
out the x axis
00:31:47.000 --> 00:31:49.500
and going in the negative x direction.
00:31:49.500 --> 00:31:51.500
So, the x value would be negative.
00:31:51.500 --> 00:31:54.500
So, the cosine would be negative, not positive.
00:31:55.000 --> 00:31:57.000
It would be exactly the same as if
00:31:57.000 --> 00:31:59.000
it was pointing this way
00:31:59.000 --> 00:32:01.000
but it is negative instead of positive.
00:32:10.000 --> 00:32:12.000
How are we doing back there?
00:32:12.000 --> 00:32:14.000
Are you getting it?
00:32:14.000 --> 00:32:16.000
Watching a video on your phone?
00:32:18.000 --> 00:32:20.000
So, how do I get the 60 degrees?
00:32:20.000 --> 00:32:22.000
This is 180 degrees.
00:32:22.000 --> 00:32:24.000
And, I want 120.
00:32:24.000 --> 00:32:26.000
So, 60 is the left over angle.
00:32:32.000 --> 00:32:34.000
Well, I was looking for sine.
00:32:36.000 --> 00:32:38.000
Another question.
00:32:44.000 --> 00:32:46.000
I am going to do a few more examples.
00:32:46.000 --> 00:32:49.000
If I do a few more you guys will get it.
00:32:58.000 --> 00:33:02.000
So, what is the answer of sine of 120? Square root of 3 over 2.
00:33:04.000 --> 00:33:06.000
And the reason is it is the same thing
00:33:06.000 --> 00:33:08.000
as the sine of 60 degrees.
00:33:13.500 --> 00:33:16.000
Cosine of 120 is negative a half.
00:33:16.000 --> 00:33:18.000
As demonstrated,
00:33:18.000 --> 00:33:20.200
cosine is negative there and sine is positive.
00:33:36.200 --> 00:33:39.200
Okay, so let us go over why these things are positive or negative.
00:33:39.200 --> 00:33:41.200
Maybe we will just use 210.
00:33:41.200 --> 00:33:43.200
Maybe that will work.
00:33:45.000 --> 00:33:47.200
How do you find the sine of 210 degrees?
00:33:47.200 --> 00:33:49.200
First, I have to find
00:33:49.200 --> 00:33:52.000
where am I going to put 210 degrees.
00:33:58.000 --> 00:34:00.000
Okay, 210 degrees
00:34:00.000 --> 00:34:02.000
is where? Well, this is 0.
00:34:02.000 --> 00:34:04.000
and this is 90.
00:34:04.000 --> 00:34:06.000
This is 180.
00:34:06.000 --> 00:34:08.000
So, 210 is somewhere
00:34:08.000 --> 00:34:10.000
down in the 3rd quadrant.
00:34:12.000 --> 00:34:14.200
Okay. If I pick a point in the 3rd quadrant.
00:34:14.200 --> 00:34:16.200
What do I know about the sine
00:34:16.200 --> 00:34:18.200
and the x and the y coordinates?
00:34:18.200 --> 00:34:20.200
Well...
00:34:21.500 --> 00:34:24.200
Any point down here is going to have a negative
00:34:24.200 --> 00:34:26.200
x-coordinate and a negative
00:34:26.200 --> 00:34:28.200
y-coordinate. So, negative x
00:34:28.500 --> 00:34:30.800
and a negative y.
00:34:32.200 --> 00:34:34.800
So, since cosine of theta is the
00:34:34.800 --> 00:34:36.800
x coordinate it is going to be
00:34:36.800 --> 00:34:38.800
negative. The sine of theta
00:34:38.800 --> 00:34:40.800
is the y coordinate. Is also
00:34:40.800 --> 00:34:43.000
going to be negative. So, in the previous problem,
00:34:43.000 --> 00:34:45.300
I was in the second quadrant.
00:34:45.300 --> 00:34:47.300
X coordinates are negative
00:34:47.300 --> 00:34:50.000
and y coordinates are positive.
00:34:50.000 --> 00:34:52.000
So, since the x coordinates are negative,
00:34:52.000 --> 00:34:54.000
cosines come out negative.
00:34:54.000 --> 00:34:56.000
Y coordinates are positive
00:34:56.200 --> 00:34:58.200
so, sines come out positive.
00:35:02.200 --> 00:35:04.200
So, I go here and I say, sine of 210
00:35:04.200 --> 00:35:06.000
well, this is 180,
00:35:06.000 --> 00:35:08.000
and this is 210,
00:35:08.000 --> 00:35:11.000
so this is 30 degrees.
00:35:12.500 --> 00:35:15.000
These all have to come out 30 and 45 and
00:35:15.000 --> 00:35:17.000
60 because we don't know any other angles.
00:35:18.000 --> 00:35:20.000
So, the sine of 210,
00:35:20.000 --> 00:35:22.000
is the same
00:35:22.000 --> 00:35:24.000
as the sine of 30 degrees.
00:35:25.700 --> 00:35:28.000
I am down here
00:35:28.000 --> 00:35:30.000
in the third quadrant.
00:35:30.000 --> 00:35:32.000
In the third quadrant, I know sine is negative.
00:35:32.000 --> 00:35:34.000
Because y coordinates are negative.
00:35:36.000 --> 00:35:40.000
So, what is the sine of 30 degrees?
00:35:40.000 --> 00:35:42.000
It is a half.
00:35:42.000 --> 00:35:44.000
This is negative a half.
00:35:45.700 --> 00:35:48.000
Do I need to go through that again?
00:35:57.600 --> 00:36:01.000
Once again, to find the sine of 210 degrees,
00:36:01.000 --> 00:36:03.000
I draw a unit circle
00:36:03.000 --> 00:36:05.000
and I go around until I have gotten
00:36:05.000 --> 00:36:07.000
to 210 degrees. This is
00:36:07.000 --> 00:36:09.000
180 that is 210
00:36:09.000 --> 00:36:12.000
so I have gone 30 degrees past the x axis.
00:36:13.000 --> 00:36:16.000
We are always going to measure angles around the x axis.
00:36:16.000 --> 00:36:18.000
Never from the y axis.
00:36:18.000 --> 00:36:20.000
Never.
00:36:20.000 --> 00:36:22.000
Okay.
00:36:22.500 --> 00:36:24.800
In the third quadrant,
00:36:24.800 --> 00:36:26.500
sines and cosines
00:36:26.500 --> 00:36:28.500
both have negative values. So,
00:36:28.500 --> 00:36:30.500
I will get exactly what I would have
00:36:30.500 --> 00:36:32.500
gotten for the sine of 30 degrees.
00:36:32.500 --> 00:36:34.500
Except it will be negative
00:36:34.500 --> 00:36:37.000
because I am pointing down instead of pointing up.
00:36:37.400 --> 00:36:39.400
Y coordinate when it is pointing up it is positive.
00:36:39.400 --> 00:36:41.400
When it is pointing down it is negative.
00:36:41.400 --> 00:36:43.400
The sine of 30 degrees
00:36:43.400 --> 00:36:45.400
this will be the same as the sine of 30 degrees.
00:36:45.400 --> 00:36:47.400
except it will be negative. It is negative a half.
00:36:48.000 --> 00:36:51.000
If I want to find the cosine of 210 degrees.
00:36:53.000 --> 00:36:55.000
It is the same angle as the sine has
00:36:55.000 --> 00:36:57.000
it will also be
00:36:57.000 --> 00:36:59.000
of 30 degrees.
00:36:59.000 --> 00:37:01.000
And, it will also be negative
00:37:01.000 --> 00:37:03.000
because it is found in the third quadrant.
00:37:05.000 --> 00:37:08.200
Cosine of 30 is the square root of 3 over 2.
00:37:10.200 --> 00:37:12.200
So, this turns out negative square root of 3 over 2.
00:37:12.200 --> 00:37:14.200
First I want to find
00:37:14.200 --> 00:37:18.000
cosine of 315 degrees.
00:37:23.700 --> 00:37:26.000
315 degrees, now where is 315 degrees?
00:37:26.000 --> 00:37:28.000
Well...
00:37:37.000 --> 00:37:39.000
Well, let us see. 270 is
00:37:39.000 --> 00:37:41.000
straight down. So,
00:37:41.000 --> 00:37:44.000
315, I am going to keep going
00:37:45.000 --> 00:37:48.000
Until I pass 270.
00:37:51.800 --> 00:37:54.200
And I get to 315 degrees.
00:37:57.200 --> 00:37:59.200
So, now what is the left over angle
00:37:59.200 --> 00:38:02.200
between this line, this radius and the x axis?
00:38:02.200 --> 00:38:04.200
45 degrees.
00:38:07.500 --> 00:38:10.200
So, the cosine of 315 will be
00:38:10.200 --> 00:38:12.200
the same as
00:38:12.200 --> 00:38:14.200
cosine of 45. Now in the 4th
00:38:14.200 --> 00:38:16.500
quadrant. Is x positive or negative?
00:38:16.500 --> 00:38:18.500
X is positive.
00:38:18.500 --> 00:38:20.500
So, you are going this way. Is Y positive or negative?
00:38:20.500 --> 00:38:22.500
Y is negative.
00:38:22.500 --> 00:38:24.300
So, cosine will be positive
00:38:24.300 --> 00:38:26.000
and sine will be negative.
00:38:26.000 --> 00:38:28.000
Because x will be positive
00:38:28.000 --> 00:38:30.000
and y will be negative.
00:38:30.000 --> 00:38:32.000
That means cosine
00:38:32.000 --> 00:38:34.000
positive
00:38:34.000 --> 00:38:36.000
Sine is negative. Okay?
00:38:37.800 --> 00:38:40.000
And what is the cosine of 45 degrees?
00:38:40.000 --> 00:38:43.000
1 over square root of 2. So this will be
00:38:44.000 --> 00:38:47.000
rad of 2 over 2.
00:38:49.500 --> 00:38:51.500
What if I wanted to find the
00:38:51.500 --> 00:38:53.500
sine of 315 degrees?
00:38:59.500 --> 00:39:02.300
Well, that is going to be the same as the sine of 45 degrees.
00:39:06.000 --> 00:39:08.600
Plus, the y coordinate is negative.
00:39:08.600 --> 00:39:10.300
So, the sine would be negative.
00:39:10.300 --> 00:39:13.000
So, this would be negative sine of 45 degrees.
00:39:13.000 --> 00:39:15.500
This is negative radical 2 over 2.
00:39:23.900 --> 00:39:25.500
How about if I did the
00:39:25.500 --> 00:39:27.500
sine of 420 degrees?
00:39:29.200 --> 00:39:31.500
420 is everyone's favorite number. (laughter)
00:39:31.500 --> 00:39:33.500
This is 360 degrees right?
00:39:33.500 --> 00:39:35.500
That is all the way around.
00:39:45.500 --> 00:39:47.500
So, I keep going another 60 degrees
00:39:47.500 --> 00:39:49.500
until I get to there.
00:39:52.700 --> 00:39:54.500
Okay. So, why is that 60?
00:39:54.500 --> 00:39:56.500
Well, I went all the way around
00:39:56.500 --> 00:39:58.500
360 and I kept going
00:39:58.500 --> 00:40:01.000
until I got to 420 degrees.
00:40:04.500 --> 00:40:06.200
I am in the first quadrant.
00:40:06.200 --> 00:40:08.200
Is sine going to be positive or negative?
00:40:08.200 --> 00:40:10.200
It is going to be positive. Because it is
00:40:10.200 --> 00:40:12.200
the y value and the y value
00:40:12.200 --> 00:40:14.200
is positive in the first quadrant.
00:40:14.200 --> 00:40:16.700
So is the x value. This will be the same thing
00:40:17.700 --> 00:40:19.700
as the sine of 60 degrees.
00:40:19.700 --> 00:40:21.700
Which is?
00:40:21.700 --> 00:40:23.700
Radical 3 over 2.
00:40:30.200 --> 00:40:32.700
Let's find the sine of 300 degrees.
00:40:37.100 --> 00:40:39.400
So, how do you find the sine of 300 degrees?
00:40:40.000 --> 00:40:42.400
Well, we have to figure out where 300 degrees is.
00:40:48.400 --> 00:40:50.400
300 degrees is
00:40:50.400 --> 00:40:52.400
not quite 360, but
00:40:52.400 --> 00:40:54.400
past 270.
00:40:56.700 --> 00:40:58.900
How much farther than 270 do we go?
00:40:58.900 --> 00:41:00.900
We go another 30 degrees.
00:41:00.900 --> 00:41:02.900
So, this angle is 60 degrees.
00:41:02.900 --> 00:41:04.900
Remember you always use the angle
00:41:04.900 --> 00:41:06.900
with the x axis. You never
00:41:06.900 --> 00:41:08.900
use the angle with the y axis.
00:41:09.400 --> 00:41:12.000
Always the x. Never the y.
00:41:14.600 --> 00:41:17.000
So, in the fourth quadrant, here,
00:41:17.000 --> 00:41:19.000
the x coordinates are positive.
00:41:19.000 --> 00:41:21.000
Y coordinates are negative.
00:41:21.000 --> 00:41:23.000
Sine is the y coordinate.
00:41:23.000 --> 00:41:25.000
So, this would be a negative value.
00:41:26.000 --> 00:41:28.000
Negative sine of 60 degrees.
00:41:29.400 --> 00:41:32.000
Sine of 60 degrees is radical 3 over 2.
00:41:40.600 --> 00:41:42.600
There is a mnemonic for this.
00:41:42.600 --> 00:41:44.600
Some of you may have learned.
00:41:55.200 --> 00:41:56.600
All students take crack.
00:41:56.600 --> 00:41:58.600
Or, All students take calculus.
00:41:58.600 --> 00:42:00.600
Or, All Seawolves,
00:42:02.600 --> 00:42:04.600
take
00:42:04.600 --> 00:42:06.600
Calculus.
00:42:08.600 --> 00:42:10.600
I learned a lot of these.
00:42:10.600 --> 00:42:13.000
Feel free to pick up whichever one you want.
00:42:17.000 --> 00:42:19.000
The goal of the mnemonic is that you
00:42:19.000 --> 00:42:21.000
remember it. That is why lots of them
00:42:21.000 --> 00:42:23.000
are obscene, because then you remember it.
00:42:24.700 --> 00:42:27.300
If they are a little off color, then you have a better chance of remembering it.
00:42:28.000 --> 00:42:30.000
What does this stand for?
00:42:30.000 --> 00:42:32.000
A is for All
00:42:32.000 --> 00:42:34.500
trig functions are positive if found in the first quadrant.
00:42:34.500 --> 00:42:36.500
So, you get an angle.
00:42:36.500 --> 00:42:39.000
You end up in the first quadrant. You get a positive answer.
00:42:39.500 --> 00:42:41.500
S stands for sine.
00:42:41.500 --> 00:42:43.500
If you are in the second quadrant
00:42:44.000 --> 00:42:46.000
sine will be positive.
00:42:46.000 --> 00:42:48.000
That means the tangent and
00:42:48.000 --> 00:42:50.000
cosine will not be positive.
00:42:53.000 --> 00:42:55.000
In the third quadrant,
00:42:55.000 --> 00:42:57.000
tangent is positive. That means
00:42:57.000 --> 00:42:59.000
sine and cosine will be negative.
00:42:59.000 --> 00:43:01.000
The tangent will be positive.
00:43:01.000 --> 00:43:03.000
The fourth quadrant, cosines are
00:43:03.000 --> 00:43:05.000
positive. The sines and
00:43:05.000 --> 00:43:07.000
tangents will be negative.
00:43:07.000 --> 00:43:10.000
So it is a very handy way to remember things.
00:43:10.000 --> 00:43:12.000
So if I asked you for,
00:43:12.000 --> 00:43:15.000
tangent of 135 degrees.
00:43:19.800 --> 00:43:21.800
Tangent is positive in the third quadrant.
00:43:22.800 --> 00:43:24.000
But, not in the fourth quadrant.
00:43:24.000 --> 00:43:25.800
Everything is positive in the first.
00:43:25.800 --> 00:43:28.000
Sine is positive in the second.
00:43:28.000 --> 00:43:30.000
Tan is positive in the third.
00:43:30.000 --> 00:43:32.000
Cosine is positive in the fourth.
00:43:33.500 --> 00:43:36.000
So, in the fourth cosine is positive, so sine
00:43:36.000 --> 00:43:38.000
and tan are negative.
00:43:41.000 --> 00:43:43.000
So, take tan of 135 degrees.
00:43:43.000 --> 00:43:45.000
So, where is 135 degrees.
00:43:45.000 --> 00:43:47.000
You don't really need to draw a circle
00:43:47.000 --> 00:43:49.000
all the time. You can.
00:43:50.000 --> 00:43:52.000
135 degrees is somewhere there.
00:43:52.000 --> 00:43:54.000
How do I know that?
00:43:54.000 --> 00:43:56.000
That's 90.
00:43:56.000 --> 00:43:58.000
That is 180.
00:43:59.000 --> 00:44:01.000
So, 135
00:44:02.500 --> 00:44:05.000
That comes out 45 degrees.
00:44:05.000 --> 00:44:07.000
Notice, practice the subtraction,
00:44:07.000 --> 00:44:09.000
but you can't get like 55 degrees.
00:44:09.000 --> 00:44:11.000
You don't know what to do with 55 degrees.
00:44:11.000 --> 00:44:13.000
It has got to be 45 degrees.
00:44:15.300 --> 00:44:17.300
So, the tangent of 135 degrees
00:44:17.300 --> 00:44:19.300
would be the same
00:44:19.500 --> 00:44:22.000
as the tangent of 45 degrees.
00:44:23.000 --> 00:44:25.200
Except, we are in the second quadrant.
00:44:25.500 --> 00:44:27.300
And, in the second quadrant,
00:44:27.300 --> 00:44:29.800
we are up there with Seawolves, okay?
00:44:29.800 --> 00:44:31.800
Only sine is positive.
00:44:31.800 --> 00:44:33.500
Tangent would be negative.
00:44:33.500 --> 00:44:35.500
So, this is negative.
00:44:36.000 --> 00:44:38.000
What is the tan of 45?
00:44:38.000 --> 00:44:40.000
The tan of 45 is 1.
00:44:40.000 --> 00:44:42.000
Well, this is negative 1.
00:44:52.000 --> 00:44:54.000
Well, these are very important to get down.
00:44:57.000 --> 00:45:00.000
Sine, cosine and
00:45:00.000 --> 00:45:02.000
tangent of
00:45:02.000 --> 00:45:04.000
315 degrees.
00:45:12.000 --> 00:45:14.300
Where is the angle at 315 degrees?
00:45:16.300 --> 00:45:18.300
315 degrees is down
00:45:18.300 --> 00:45:20.300
here.
00:45:20.300 --> 00:45:22.300
It gets kind of boring drawing that circle all the time.
00:45:22.300 --> 00:45:24.300
So I am not going to draw it.
00:45:24.300 --> 00:45:26.300
So here the angle is down here.
00:45:28.300 --> 00:45:30.300
Because that is 360
00:45:30.300 --> 00:45:32.300
and that is 270.
00:45:32.300 --> 00:45:34.300
So what is my left over angle?
00:45:34.300 --> 00:45:36.300
45 degrees.
00:45:40.300 --> 00:45:42.300
Okay, so this will be the same
00:45:42.300 --> 00:45:44.300
so sine of 315
00:45:44.300 --> 00:45:46.300
will be the same
00:45:46.300 --> 00:45:48.300
as sine of 45, except
00:45:48.300 --> 00:45:50.300
we are in quadrant, the fourth quadrant.
00:45:50.300 --> 00:45:52.300
only cosine is positive.
00:45:56.300 --> 00:45:59.300
All of these will be at the 45 degree angle.
00:45:59.300 --> 00:46:01.300
But, only cosine will be positive.
00:46:01.300 --> 00:46:03.300
These two will be negative.
00:46:06.500 --> 00:46:08.300
What is the sine of 45?
00:46:08.300 --> 00:46:10.300
radical 2 over 2.
00:46:10.300 --> 00:46:12.300
So this is negative radical 2 over 2.
00:46:13.000 --> 00:46:15.300
Cosine of 45 is radical 2 over 2.
00:46:15.300 --> 00:46:17.300
This is positive.
00:46:18.600 --> 00:46:21.600
Tangent would be negative 1.
00:46:23.600 --> 00:46:25.300
You got that one?
00:46:25.300 --> 00:46:27.600
Alright, it is close enough to 6:50.