WEBVTT
Kind: captions
Language: en
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So, if I told you that sine of v is
00:00:04.000 --> 00:00:08.000
3/11 and I gave you either a or c
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this would not be very hard.
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Because then what you would do is set
00:00:12.000 --> 00:00:14.000
the ratio. You would say
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3 is to 11 as this is to this.
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In fact we know that the sine of v is
00:00:22.000 --> 00:00:24.000
is a over c. So,
00:00:24.000 --> 00:00:26.000
three elevenths...3 is to 11
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as a is to c.
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The problem is that you do not know a and you do not know c.
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Right, so that is a bit of an issue.
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So, how long, what could we do?
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We have 2 variables.
00:00:36.500 --> 00:00:39.000
We could use the Pythagoren Theorem
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to get another equation.
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You also know
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That a squared plus 4 squared
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equals c squared.
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So, now we have to do a kind of nightmarish
00:00:51.500 --> 00:00:53.500
algebra that's
00:00:53.500 --> 00:00:55.500
the reason we will figure it out.
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Um, this is the
00:00:59.500 --> 00:01:01.500
same as a squared plus 16
00:01:01.500 --> 00:01:04.000
equals c squared.
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And if you cross multiply,
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3 c is 11 a.
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So, we can do a little substituting,
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Okay?
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and solve it. We are trying to find what a is
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so we want to get rid of c.
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C is 11 a over 3.
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Now, we plug it into here.
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So, a squared
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plus 16
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is (11 over 3 a) squared.
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Alright, lets solve that.
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A squared plus 16
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equals 11 squared, which is 121,
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a squared over 9.
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Another board...
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So, we have...
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a squared plus 16
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is 121 a squared over 9
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We all love fractions.
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The best way to deal with a fraction
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is to multiply everything by 9.
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Now you get...
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9 a squared plus 144 is
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121 a squared.
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You are sitting here saying that I would never do this on an exam.
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I have faith.
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Subtract the 9 a squared..
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You get 144 is
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112 a squared.
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Divide it by 112.
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Then you take the square root of it.
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a is
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the square root
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of 144 over 112.
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So,
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did I make a mistake?
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Sure. Let's review.
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We tell you that sine v is
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3/11. And, we are trying to find a.
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So, you go to your triangle
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and you say that you know that the
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sine of v is the opposite
00:03:01.200 --> 00:03:03.200
over the hypotenuse.
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That means that 3/11 is
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the same as a over c.
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And, as I said before, the problem is
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there is 2 variables and we only have one equation.
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We need another equation.
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You say, "I don't know another equation."
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"Except, I love the Pythagorean Theorem. It is the only one I ever memorized."
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a squared plus 4 squared equals c squared.
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If a is 3
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right?! and c is 11,
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3 squared plus 4 squared is not going to give you 11 squared.
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That is the problem.
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Okay?
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Um, so, now
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we have a second equation.
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Now you just have to do messy algebra.
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So, the way that you solve this is that you could
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isolate a or c in this one to get something squared.
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That is what you want to do.
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So in this equation, you cross multiply
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and isolate c
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and you call it 11 a over 3.
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And, you place it here for c.
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Then you square that.
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As I said, you multiply through by 9
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and you get this nice, not too hard equation.
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Subtract the 9.
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Confused?
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Another
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thing that we seem to have this
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TA versus Professor conflict, is
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when we give you something like,
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theta in between
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Pi over 2
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and Pi
00:04:29.000 --> 00:04:31.000
sin
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theta is
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4/7
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find cosine theta
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We did some of these one Monday, but let us do one more.
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Okay? There is more than one
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way to get this right. What you want
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to do is to find the most straight forward way
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to get this right.
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The first thing is where is Pi over 2?
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Well, Pi is
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180 degrees so that is 90 degrees.
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That is 180 degrees
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So it is somewhere in the second quadrant.
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Remember what I said, "First you need to figure out where to
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place the angle.
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Now, the sine is 4 over 7.
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do Pythagorean Theorem of 4 and 7.
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You could do some other squaring
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and Pythagorean stuff but there is no need to.
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We need to find,
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That side because we need to
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find the cosine of theta.
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Okay? So, 4 squared
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plus b squared is 7 squared.
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Okay, 16 plus b squared,
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is 49.
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b squared is 33.
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b is plus or minus the square root of 33.
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We are in the second quadrant
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so be is in this direction. So, b is going
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to be a negative value. So, b is going to be
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negative square root of 33.
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So, therefore, cosine of theta
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is SOH CAH TOA.
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Negative square root of 33
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over 7.
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They are going to capture it on video.
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I want to keep that on the DL.
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Okay. So, some other stuff
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you should know.
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Sine of
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a negative angle
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is negative sine of a positive angle.
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What does that mean?
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What does the negative angle mean?
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Well, this is the angle, theta. Right?
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You go this way.
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Negative theta just means go the other way.
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So, that is the corresponding
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negative angle. So, if this is
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40 degrees
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this is negative 40 degrees. If this is
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Pi over 5, this is negative Pi over 5.
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So, the sine of
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positive theta is
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that y.So, the sine of negative theta
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is going to be negative y.
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If you know that the sine of
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17 degrees
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is point 2.
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Then the sine of negative 17 degrees
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is negative point 2.
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Okay. Not very hard.
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The cosine of a
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negative angle is the
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same
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as the cosine of a positive angle.
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Because they have the same x.
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So the cosine of this
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angle is the same
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as the cosine of the negative angle.
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Okay? So this is what is called an odd function,
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and this is an even function.
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Later we will do odd and even functions
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So, that is one thing you should know.
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Will that show up in a problem? Well,
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the problem would be using the sine of a negative
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angle in it. And we would convert it
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negative sine of a positive angle.
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That is all that we would be testing.
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Another thing is this
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angle is theta
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and since the sum of these two angles is 90 degrees
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or Pi over 2.This angle is
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Pi over 2 minus theta.
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So, I pointed this out
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last week.
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The sine of the one angle is the cosine of the other angle.
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Right, this is A, B, and C.
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The sine of theta
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is A over C.
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Cosine of
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Pi over 2 minus theta.
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Is also A over C.
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That tells you that the sine of
00:09:08.500 --> 00:09:10.500
theta
00:09:10.500 --> 00:09:12.500
is cosine
00:09:13.000 --> 00:09:15.000
of Pi over 2 minus theta.
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Also, known as 90 degrees minus theta.
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That
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means that they are complementary angles.
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2 angles add up to 90 degrees.
00:09:24.600 --> 00:09:26.600
or Pi over 2 ok?
00:09:26.600 --> 00:09:28.600
So, if the sine of the one angle
00:09:28.600 --> 00:09:30.600
is the cosine of the other angle.
00:09:30.600 --> 00:09:32.000
And, of course
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cosine of theta
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is sine of pi over 2 minus theta.
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Suppose that I tell you...
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sine of 2x-5 equals
00:09:54.000 --> 00:09:55.700
equals 0.
00:09:55.700 --> 00:09:58.700
Find the smallest x where that is true.
00:09:58.700 --> 00:10:00.700
Smallest positive x.
00:10:00.700 --> 00:10:02.700
00:10:02.700 --> 00:10:04.700
00:10:04.700 --> 00:10:06.700
00:10:06.700 --> 00:10:08.700
00:10:08.700 --> 00:10:10.700
00:10:10.700 --> 00:10:12.700
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Well,
00:10:16.400 --> 00:10:18.400
for sine of 2x-5 equals
00:10:18.400 --> 00:10:21.000
0 sine 0 is 0.
00:10:21.000 --> 00:10:23.000
We learned that one. So this just means
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2x-5=0.
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2x=5
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x=5/2
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Cosine of 2x-5=0
00:10:36.000 --> 00:10:38.000
00:10:38.000 --> 00:10:40.900
Where is cosine equal to 0?
00:10:40.900 --> 00:10:42.900
00:10:42.900 --> 00:10:44.900
90
00:10:44.900 --> 00:10:46.900
also known as Pi over 2.
00:10:46.900 --> 00:10:49.200
So, when you do these problems, you always do them in radians.
00:10:50.200 --> 00:10:52.200
So, the cosine of 90 degrees,
00:10:52.200 --> 00:10:54.200
the cosine of Pi over 2 is 0.
00:10:54.200 --> 00:10:56.200
So, 2x-5 has to equal Pi over 2.
00:10:56.200 --> 00:10:58.200
We can just solve that for x.
00:10:58.200 --> 00:11:00.200
So you add 5.
00:11:00.200 --> 00:11:02.200
00:11:02.200 --> 00:11:04.200
00:11:04.200 --> 00:11:06.200
and you divide it by 2.
00:11:06.200 --> 00:11:08.200
Pi over 2 plus 5
00:11:09.200 --> 00:11:12.200
divided by 2. And, you could leave it like that.
00:11:12.200 --> 00:11:16.200
Okay? You do not need to simplify unless we ask.
00:11:16.200 --> 00:11:18.200
So, 2x-5 = Pi over 2.
00:11:18.200 --> 00:11:20.200
So, to solve that you just add 5 to
00:11:20.200 --> 00:11:22.200
both sides
00:11:22.200 --> 00:11:24.200
00:11:24.200 --> 00:11:26.200
and divide by 2.
00:11:26.200 --> 00:11:28.200
00:11:29.200 --> 00:11:32.200
So, these are all in terms of radians.
00:11:32.200 --> 00:11:34.200
00:11:34.200 --> 00:11:36.200
Ah, I said the smallest x.
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Smallest positive x where it is true.
00:11:38.200 --> 00:11:40.200
Now, yes, you could use 3 Pi over 2.
00:11:40.200 --> 00:11:42.200
You could use 5 Pi over 2. There is
00:11:42.200 --> 00:11:45.000
an infinite number of answers. So, we want the smallest one.
00:11:45.000 --> 00:11:47.000
00:11:47.000 --> 00:11:49.000
00:11:49.000 --> 00:11:51.000
00:11:51.000 --> 00:11:53.000
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00:11:55.000 --> 00:11:57.000
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00:12:01.000 --> 00:12:03.000
00:12:03.800 --> 00:12:05.800
How about the smallest positive x where
00:12:05.800 --> 00:12:07.300
this is true?
00:12:07.300 --> 00:12:09.300
Lets see if you could figure it out.
00:12:09.300 --> 00:12:11.300
00:12:12.800 --> 00:12:15.300
Where is sine equal to 1/2?
00:12:15.300 --> 00:12:17.300
Lots of places plus
00:12:17.300 --> 00:12:19.300
00:12:19.300 --> 00:12:21.300
not degrees
00:12:21.300 --> 00:12:23.300
Pi over 6.
00:12:23.300 --> 00:12:25.300
Do not do it in degrees, you have to do it in radians.
00:12:25.300 --> 00:12:27.300
So this means
00:12:27.300 --> 00:12:30.000
the sine of Pi over 6 is 1/2.
00:12:31.300 --> 00:12:33.300
00:12:34.300 --> 00:12:36.000
2x-1
00:12:36.000 --> 00:12:38.000
will give you Pi over 6.
00:12:38.000 --> 00:12:40.000
00:12:40.000 --> 00:12:42.000
Now you need to figure out what x is
00:12:42.000 --> 00:12:44.000
so, add 1
00:12:44.000 --> 00:12:46.000
00:12:46.000 --> 00:12:48.000
and divide by 2.
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00:12:50.000 --> 00:12:52.000
00:12:52.000 --> 00:12:54.000
00:12:54.000 --> 00:12:56.000
How about that one?
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The smallest positive x where
00:12:58.000 --> 00:13:00.000
that is true. Tangent of 4x
00:13:00.000 --> 00:13:02.000
minus 3
00:13:02.000 --> 00:13:04.000
is 1.
00:13:04.000 --> 00:13:06.000
00:13:06.000 --> 00:13:08.000
00:13:08.000 --> 00:13:10.000
00:13:10.000 --> 00:13:12.000
00:13:12.000 --> 00:13:14.000
So, you have to say to yourself,
00:13:14.000 --> 00:13:16.000
where is tangent equal to 1?
00:13:16.000 --> 00:13:18.000
Lots of places, but the
00:13:18.000 --> 00:13:20.000
smallest positive value that is equal to 1
00:13:20.000 --> 00:13:22.000
is Pi over 4.
00:13:22.000 --> 00:13:24.000
00:13:24.000 --> 00:13:26.000
00:13:26.000 --> 00:13:28.000
Okay?
00:13:28.000 --> 00:13:30.000
So, now...
00:13:30.000 --> 00:13:33.000
you just solve for x, and you add 3.
00:13:33.000 --> 00:13:35.000
00:13:35.000 --> 00:13:37.000
00:13:37.000 --> 00:13:39.000
and divide by 4.
00:13:39.000 --> 00:13:41.000
00:13:41.000 --> 00:13:43.000
00:13:43.000 --> 00:13:45.000
You can leave it like that.
00:13:45.000 --> 00:13:47.000
Do not
00:13:47.000 --> 00:13:49.000
00:13:50.000 --> 00:13:53.000
do not write that.
00:13:53.000 --> 00:13:55.000
The problem is that it looks like you are writing
00:13:55.000 --> 00:13:57.000
3/4.
00:13:57.000 --> 00:13:59.000
So, if you want to use that slash
00:13:59.000 --> 00:14:01.000
make sure you put parentheses around the
00:14:01.000 --> 00:14:03.000
Pi over 4 plus 3. I want to recommend
00:14:03.000 --> 00:14:05.000
not doing it that way.
00:14:05.000 --> 00:14:07.000
Okay?
00:14:07.000 --> 00:14:09.000
00:14:09.000 --> 00:14:11.000
00:14:11.500 --> 00:14:13.500
So what if I asked you that?
00:14:13.500 --> 00:14:15.000
00:14:15.000 --> 00:14:17.500
What if I was in a really bad mood? So, I
00:14:17.500 --> 00:14:19.500
said, "How about that one?"
00:14:19.500 --> 00:14:21.500
00:14:21.500 --> 00:14:23.500
00:14:23.500 --> 00:14:25.500
00:14:25.500 --> 00:14:27.500
It is not as bad as it looks.
00:14:27.500 --> 00:14:29.500
Because remember just a couple minutes ago,
00:14:29.500 --> 00:14:31.500
I showed you an identity.
00:14:31.500 --> 00:14:33.500
What do you know about the relationship
00:14:33.500 --> 00:14:35.500
between the sine of an angle
00:14:35.500 --> 00:14:37.500
and cosine? We know that
00:14:37.500 --> 00:14:39.500
sine of the one angle is the cosine of
00:14:39.500 --> 00:14:41.500
the complementary angle.
00:14:41.500 --> 00:14:43.500
So, cosine of x is the same as
00:14:43.500 --> 00:14:45.500
the sine of
00:14:45.500 --> 00:14:47.500
Pi over 2 minus x.
00:14:47.500 --> 00:14:49.500
Okay?
00:14:49.500 --> 00:14:51.500
So, we could say that the sine of
00:14:51.500 --> 00:14:53.500
2x-1
00:14:53.500 --> 00:14:55.500
is the same as the sine of
00:14:55.500 --> 00:14:57.500
Pi over 2 minus x.
00:14:57.500 --> 00:14:59.500
So, remember, the cosine of
00:14:59.500 --> 00:15:01.500
the angle is the same as the sine of Pi over 2
00:15:01.500 --> 00:15:03.500
minus the same angle.
00:15:05.000 --> 00:15:07.500
Now, I just say 2x-1
00:15:07.500 --> 00:15:09.500
is Pi over 2 minus x.
00:15:09.500 --> 00:15:11.500
00:15:11.500 --> 00:15:13.500
Yes?
00:15:13.500 --> 00:15:15.500
00:15:15.500 --> 00:15:17.500
00:15:17.500 --> 00:15:19.500
This is x
00:15:19.500 --> 00:15:21.500
that angle is
00:15:21.500 --> 00:15:23.500
Pi over 2 minus x.
00:15:23.500 --> 00:15:25.500
Sine of this angle
00:15:25.500 --> 00:15:27.200
is the cosine of that angle.
00:15:27.200 --> 00:15:29.500
The cosine of this angle is the sine of that angle.
00:15:29.500 --> 00:15:31.500
00:15:31.500 --> 00:15:33.500
00:15:33.500 --> 00:15:35.500
00:15:35.500 --> 00:15:37.500
So, sine of x is
00:15:37.500 --> 00:15:39.500
B/C
00:15:44.000 --> 00:15:46.000
Cosine is Pi over 2
00:15:46.000 --> 00:15:48.000
minus x
00:15:48.000 --> 00:15:50.300
is also B/C
00:15:50.300 --> 00:15:52.300
Therefore the sine of x
00:15:52.300 --> 00:15:54.300
is the cosine of Pi over 2 minus x.
00:15:54.300 --> 00:15:56.300
00:15:57.000 --> 00:15:59.000
That is where that came from before.
00:16:00.000 --> 00:16:03.000
And, the cosine of x is the sine of Pi over 2 minus x.
00:16:03.000 --> 00:16:05.000
00:16:05.000 --> 00:16:07.000
But, if we give you something like this
00:16:07.000 --> 00:16:09.000
This is your typical Webassign homework
00:16:09.000 --> 00:16:11.000
meaning that this caliber of problem.
00:16:11.000 --> 00:16:13.000
If you just replace the cosine
00:16:13.000 --> 00:16:15.000
with sine of Pi over 2 minus x.
00:16:15.000 --> 00:16:17.000
The sine of this is the sine of that
00:16:17.000 --> 00:16:19.000
and you can pull away your sines.
00:16:19.000 --> 00:16:21.000
So, 2x-1 is just equal to Pi over 2 minus x.
00:16:22.000 --> 00:16:24.000
Add x to both sides.
00:16:24.000 --> 00:16:26.000
Add 1 to both sides.
00:16:26.000 --> 00:16:28.000
x equals Pi over 2 plus 1 all over 3.
00:16:28.000 --> 00:16:30.000
00:16:30.000 --> 00:16:32.000
00:16:32.000 --> 00:16:34.000
00:16:34.000 --> 00:16:36.000
00:16:36.000 --> 00:16:38.000
00:16:38.000 --> 00:16:40.000
00:16:40.000 --> 00:16:42.000
00:16:42.000 --> 00:16:44.000
00:16:44.000 --> 00:16:46.000
This should take you a minute.
00:16:46.000 --> 00:16:48.000
So, if you want to turn this into cosine.
00:16:48.000 --> 00:16:50.000
I could turn this into sine or
00:16:50.000 --> 00:16:52.000
I could turn this into cosine.
00:16:52.000 --> 00:16:54.000
00:16:54.000 --> 00:16:56.000
It would be cosine of Pi over 2
00:16:56.000 --> 00:16:58.000
minus whatever
00:16:58.000 --> 00:17:00.000
this angle is. So, this angle
00:17:00.000 --> 00:17:02.000
is x minus 2.
00:17:02.000 --> 00:17:04.000
00:17:04.000 --> 00:17:06.000
00:17:06.000 --> 00:17:08.000
Be very careful when
00:17:08.000 --> 00:17:10.000
you do this it should be the minus theta here.
00:17:10.000 --> 00:17:12.000
This is the angle, this is the theta.
00:17:12.000 --> 00:17:14.000
So this is
00:17:14.000 --> 00:17:16.000
the Pi over 2 minus
00:17:16.000 --> 00:17:18.000
theta. Okay?
00:17:18.000 --> 00:17:20.000
You sure? I see
00:17:20.000 --> 00:17:22.000
a lot of people confused on this.
00:17:22.000 --> 00:17:24.000
00:17:24.000 --> 00:17:26.000
You have to put it in parentheses because it
00:17:26.000 --> 00:17:28.000
would otherwise mess up your minus sign.
00:17:28.000 --> 00:17:30.000
Maybe you would not mess up the minus sign,
00:17:30.000 --> 00:17:32.000
but most people will.
00:17:32.000 --> 00:17:34.000
This means 4x-5
00:17:34.000 --> 00:17:36.000
equals
00:17:36.000 --> 00:17:38.000
Pi over 2
00:17:38.000 --> 00:17:40.000
minus (x-2)
00:17:40.000 --> 00:17:42.000
00:17:42.000 --> 00:17:44.000
So, 4x-5
00:17:44.000 --> 00:17:46.000
is Pi over 2 minus
00:17:46.000 --> 00:17:48.000
whatever that angle is.
00:17:48.000 --> 00:17:50.000
Okay?
00:17:50.000 --> 00:17:52.000
This becomes 4x-5
00:17:52.000 --> 00:17:54.000
00:17:54.000 --> 00:17:56.000
equals Pi over 2
00:17:56.000 --> 00:17:58.000
minus x plus 2.
00:17:58.000 --> 00:18:00.000
00:18:00.000 --> 00:18:02.000
00:18:02.000 --> 00:18:04.000
So, 5x
00:18:04.000 --> 00:18:06.000
equals
00:18:06.000 --> 00:18:08.000
Pi over 2
00:18:08.000 --> 00:18:10.000
plus 7.
00:18:10.000 --> 00:18:12.000
Because you divide by 5.
00:18:12.000 --> 00:18:14.000
Okay?
00:18:14.000 --> 00:18:16.000
x equals
00:18:16.000 --> 00:18:18.000
Pi over 2
00:18:18.000 --> 00:18:20.000
plus 7.
00:18:20.000 --> 00:18:22.000
All over Pi.
00:18:22.000 --> 00:18:24.000
Yes.
00:18:24.000 --> 00:18:26.000
00:18:26.000 --> 00:18:28.000
It is the same thing. Yes.
00:18:28.000 --> 00:18:30.000
Not 90 degrees though.
00:18:30.000 --> 00:18:32.000
It must be Pi over 2.
00:18:32.000 --> 00:18:34.000
00:18:34.000 --> 00:18:36.000
00:18:36.000 --> 00:18:38.000
Oh, you also
00:18:38.000 --> 00:18:40.000
could have made this into sine.
00:18:40.000 --> 00:18:42.000
Okay? You could have said sine of x
00:18:42.000 --> 00:18:44.000
minus 2 is the sine of Pi
00:18:44.000 --> 00:18:46.000
of 2 minus 4x-5.
00:18:46.000 --> 00:18:48.000
In the end you get the same angle.
00:18:48.000 --> 00:18:50.000
00:18:50.000 --> 00:18:52.000
00:18:52.000 --> 00:18:54.000
00:18:54.000 --> 00:18:56.000
00:18:56.000 --> 00:18:58.000
00:18:58.000 --> 00:19:00.000
00:19:00.000 --> 00:19:02.000
00:19:02.000 --> 00:19:04.000
00:19:04.000 --> 00:19:06.000
00:19:06.000 --> 00:19:08.000
00:19:08.000 --> 00:19:10.000
You can convert this to
00:19:10.000 --> 00:19:12.000
sine or you can convert this
00:19:12.000 --> 00:19:14.000
to cosine. It doesn't matter.
00:19:14.000 --> 00:19:16.000
Okay? So, this
00:19:16.000 --> 00:19:18.000
is going to be the same as sine
00:19:18.000 --> 00:19:20.000
of Pi over 2
00:19:20.000 --> 00:19:22.000
minus
00:19:22.000 --> 00:19:24.000
2-x.
00:19:24.000 --> 00:19:26.000
00:19:26.000 --> 00:19:28.000
00:19:28.000 --> 00:19:30.000
00:19:30.000 --> 00:19:32.000
00:19:32.000 --> 00:19:34.000
00:19:34.000 --> 00:19:36.000
Distribute the minus sign.
00:19:36.000 --> 00:19:38.000
Well, you can throw away the sine.
00:19:38.000 --> 00:19:40.000
And you can say, 3x+
00:19:40.000 --> 00:19:42.000
4 is Pi over 2
00:19:42.000 --> 00:19:45.000
minus 2
00:19:45.000 --> 00:19:47.000
00:19:47.000 --> 00:19:49.500
plus x.
00:19:49.500 --> 00:19:51.500
You must get that minus sign in there.
00:19:51.500 --> 00:19:53.500
Pi over 2 minus
00:19:53.500 --> 00:19:55.500
whatever that angle is.
00:19:55.500 --> 00:19:57.500
00:19:57.500 --> 00:19:59.500
00:19:59.500 --> 00:20:01.500
00:20:01.500 --> 00:20:03.500
Subtract x from both sides.
00:20:03.500 --> 00:20:05.500
Subtract 4 from both
00:20:05.500 --> 00:20:07.500
sides.
00:20:07.500 --> 00:20:09.500
Divide by 2.
00:20:09.500 --> 00:20:11.500
00:20:11.500 --> 00:20:13.500
00:20:13.500 --> 00:20:15.500
00:20:15.500 --> 00:20:17.500
00:20:17.500 --> 00:20:19.500
Okay?
00:20:19.500 --> 00:20:21.500
Feeling happier?
00:20:21.500 --> 00:20:23.500
Good.
00:20:23.500 --> 00:20:25.500
So, let us see, now, what these would be.
00:20:25.500 --> 00:20:27.500
00:20:27.500 --> 00:20:29.500
Sine of theta plus Pi.
00:20:29.500 --> 00:20:31.500
Cosine of theta plus Pi.
00:20:31.500 --> 00:20:33.500
00:20:33.500 --> 00:20:35.500
What does that mean to be plus Pi?
00:20:35.500 --> 00:20:37.500
Well, let us think about the unit circle for a second.
00:20:37.500 --> 00:20:39.500
00:20:39.500 --> 00:20:41.500
If that is theta.
00:20:41.500 --> 00:20:43.500
If you are going Pi
00:20:43.500 --> 00:20:45.500
it is going 180 degrees.
00:20:45.500 --> 00:20:47.500
So that is coming around to the other side.
00:20:47.500 --> 00:20:49.500
00:20:49.500 --> 00:20:51.500
00:20:51.500 --> 00:20:53.500
This would be theta plus Pi.
00:20:54.000 --> 00:20:57.000
I went Pi. I went half way around the circle.
00:20:57.000 --> 00:20:59.000
So, sine
00:20:59.000 --> 00:21:01.000
here is the negative sine
00:21:01.000 --> 00:21:03.000
here. The cosine
00:21:03.000 --> 00:21:05.000
here is the negative cosine
00:21:05.000 --> 00:21:07.000
here.
00:21:07.000 --> 00:21:09.000
So, by doing theta plus Pi
00:21:09.000 --> 00:21:11.000
I am switching the signs
00:21:11.000 --> 00:21:13.000
from positive to negative or from
00:21:13.000 --> 00:21:15.000
negative to positive. So, again this angle
00:21:15.000 --> 00:21:17.000
theta. When I went
00:21:17.000 --> 00:21:19.000
this far.
00:21:19.000 --> 00:21:21.000
Then, adding another Pi means I am going
00:21:21.000 --> 00:21:23.000
half way around the circle. I am doing 180 degrees.
00:21:24.300 --> 00:21:27.000
That means now I am down here.
00:21:27.000 --> 00:21:28.500
00:21:28.500 --> 00:21:31.000
So, all that happens is values
00:21:31.000 --> 00:21:33.000
that were positive here, become negative here.
00:21:33.000 --> 00:21:35.000
and values that become positive here were negative here.
00:21:35.000 --> 00:21:37.000
And, the other way around.
00:21:37.000 --> 00:21:39.000
00:21:39.000 --> 00:21:41.000
00:21:41.000 --> 00:21:43.000
00:21:43.000 --> 00:21:45.000
00:21:45.000 --> 00:21:47.000
00:21:47.000 --> 00:21:49.000
00:21:49.000 --> 00:21:51.000
00:21:51.000 --> 00:21:53.000
00:21:53.000 --> 00:21:55.000
And, of course
00:21:55.000 --> 00:21:57.000
if I add 2 Pi,
00:21:57.000 --> 00:21:59.000
I get back where I started.
00:21:59.000 --> 00:22:01.000
00:22:01.000 --> 00:22:03.000
00:22:03.000 --> 00:22:05.000
00:22:05.000 --> 00:22:08.000
So, what does 2 Pi mean? It means that I went all the way around the circle.
00:22:08.000 --> 00:22:10.000
00:22:10.000 --> 00:22:12.000
00:22:12.000 --> 00:22:14.000
00:22:14.000 --> 00:22:16.000
00:22:16.000 --> 00:22:18.000
In fact when we have 2 Pi,
00:22:18.000 --> 00:22:20.000
we are back where we started and everything starts
00:22:20.000 --> 00:22:22.000
repeating. So that is called a
00:22:22.000 --> 00:22:24.000
period.
00:22:24.000 --> 00:22:26.000
A circle is one full period and it starts over.
00:22:26.000 --> 00:22:28.000
00:22:28.000 --> 00:22:30.000
00:22:30.000 --> 00:22:32.000
00:22:32.000 --> 00:22:34.000
00:22:34.000 --> 00:22:36.000
So, where is theta plus 2 Pi?
00:22:36.000 --> 00:22:38.000
Well, that is theta,
00:22:38.000 --> 00:22:40.000
that is theta plus 2 Pi. I went one
00:22:40.000 --> 00:22:42.000
full cycle around.
00:22:42.000 --> 00:22:44.000
00:22:44.000 --> 00:22:46.000
00:22:46.000 --> 00:22:48.000
In fact, anytime
00:22:48.000 --> 00:22:50.000
I go Pi.
00:22:50.000 --> 00:22:52.000
I change the signs
00:22:52.000 --> 00:22:54.200
from positive to negative and negative to positive.
00:22:54.200 --> 00:22:56.200
Anytime I go 2 Pi,
00:22:56.200 --> 00:22:58.200
I get back where I started,
00:22:58.200 --> 00:23:00.200
so,
00:23:00.200 --> 00:23:02.200
If I take the
00:23:02.200 --> 00:23:04.200
sine of theta and I
00:23:04.200 --> 00:23:06.800
add k time Pi.
00:23:06.800 --> 00:23:08.800
Where k is an integer.
00:23:08.800 --> 00:23:10.800
00:23:10.800 --> 00:23:12.800
So, I have sine of theta
00:23:12.800 --> 00:23:14.800
plus k Pi
00:23:14.800 --> 00:23:16.000
00:23:16.000 --> 00:23:18.000
k is an integer.
00:23:18.000 --> 00:23:20.000
Then from that you pull,
00:23:20.500 --> 00:23:22.500
Sine is theta
00:23:24.000 --> 00:23:25.000
if
00:23:25.000 --> 00:23:28.000
k is even
00:23:28.000 --> 00:23:30.000
or negative sine
00:23:30.000 --> 00:23:32.000
of theta
00:23:32.000 --> 00:23:34.000
if k is odd.
00:23:34.000 --> 00:23:36.000
00:23:36.000 --> 00:23:38.000
00:23:38.000 --> 00:23:40.000
00:23:40.000 --> 00:23:42.000
00:23:42.000 --> 00:23:44.000
00:23:44.000 --> 00:23:46.000
Similarly,
00:23:47.000 --> 00:23:49.000
If I have cosine of theta
00:23:49.000 --> 00:23:51.000
plus k pi
00:23:51.000 --> 00:23:53.500
that will be
00:23:53.500 --> 00:23:55.500
cosine of theta
00:23:56.000 --> 00:23:58.000
when theta is even.
00:23:58.000 --> 00:24:00.000
00:24:00.000 --> 00:24:02.000
negative cosine of theta
00:24:02.000 --> 00:24:04.000
00:24:04.000 --> 00:24:06.000
if k is odd.
00:24:06.000 --> 00:24:08.000
00:24:08.000 --> 00:24:10.000
00:24:10.000 --> 00:24:12.000
00:24:13.800 --> 00:24:16.000
So, this helps us figure out something like.
00:24:16.000 --> 00:24:18.000
00:24:18.000 --> 00:24:20.000
sine of
00:24:20.000 --> 00:24:22.000
13 Pi over 6.
00:24:22.000 --> 00:24:24.000
00:24:24.000 --> 00:24:26.000
One thing you need to do of course
00:24:26.000 --> 00:24:28.000
is to turn that into degrees.
00:24:28.000 --> 00:24:30.000
Another thing you can do of course is say
00:24:31.000 --> 00:24:33.000
that is Pi over 6
00:24:33.000 --> 00:24:35.000
plus 12 Pi over 6.
00:24:35.000 --> 00:24:37.000
00:24:37.000 --> 00:24:39.000
Why do I need the 12 Pi over 6?
00:24:39.000 --> 00:24:41.000
Because that is 2 Pi.
00:24:41.000 --> 00:24:43.000
00:24:44.000 --> 00:24:46.000
That is the same thing as sine
00:24:46.000 --> 00:24:48.000
of Pi over 6
00:24:48.500 --> 00:24:50.500
Plus, 2 Pi.
00:24:50.500 --> 00:24:52.500
00:24:52.500 --> 00:24:54.500
00:24:54.500 --> 00:24:56.500
00:24:56.500 --> 00:24:58.500
00:24:58.500 --> 00:25:00.500
And we know that if i add 2 Pi,
00:25:00.500 --> 00:25:02.500
and because 2 is an even number
00:25:02.500 --> 00:25:04.500
that is just the same
00:25:04.500 --> 00:25:06.500
as sine of Pi over 6.
00:25:07.300 --> 00:25:09.300
Did we all memorize our charts yet?
00:25:09.300 --> 00:25:11.300
NO?!
00:25:11.300 --> 00:25:13.300
You should get them tattooed.
00:25:13.300 --> 00:25:15.300
00:25:16.000 --> 00:25:18.000
Tattoo it right about here.
00:25:18.000 --> 00:25:20.000
So, you can cross your leg in the test.
00:25:20.000 --> 00:25:22.000
You can look down and "Oh!"
00:25:22.000 --> 00:25:24.000
00:25:24.000 --> 00:25:26.000
00:25:26.000 --> 00:25:28.000
It is equal to a half.
00:25:28.000 --> 00:25:30.000
00:25:30.000 --> 00:25:32.000
00:25:32.000 --> 00:25:35.000
If I give you this, you should be able to convert it to degrees.
00:25:35.000 --> 00:25:37.000
00:25:37.000 --> 00:25:39.000
We should try to figure out
00:25:39.000 --> 00:25:41.500
how many 2 Pis am I adding to this?
00:25:42.200 --> 00:25:44.200
That is another way to figure it out.
00:25:44.200 --> 00:25:46.200
So, one thing you can do to make your life easier,
00:25:46.200 --> 00:25:48.200
just sort of start subtracting
00:25:48.200 --> 00:25:50.200
2 Pis
00:25:50.200 --> 00:25:52.200
Why 2 Pis?
00:25:52.200 --> 00:25:54.800
Anytime I add 2 Pi I go once around the circle.
00:25:54.800 --> 00:25:56.800
Okay? So if I say,
00:25:56.800 --> 00:25:58.800
how many times around the circle do I go?
00:25:58.800 --> 00:26:00.800
Well, let's subtract 2 Pi.
00:26:00.800 --> 00:26:02.200
00:26:02.200 --> 00:26:04.200
That is 17 Pi over 4 minus
00:26:04.200 --> 00:26:06.200
8 Pi over 4.
00:26:06.200 --> 00:26:08.000
00:26:08.000 --> 00:26:10.200
Which is 9 Pi over 4.
00:26:10.200 --> 00:26:12.200
Okay?
00:26:12.200 --> 00:26:14.200
So,
00:26:14.200 --> 00:26:16.200
sure.
00:26:16.200 --> 00:26:18.200
17 Pi over 4 is where?
00:26:18.200 --> 00:26:20.200
Well, I have to start
00:26:20.200 --> 00:26:22.200
somewhere and
00:26:22.200 --> 00:26:24.200
it is bigger than 2 Pi, so I must have gone around the circle
00:26:24.200 --> 00:26:26.200
more than once.
00:26:26.200 --> 00:26:28.200
The question is how far did I go?
00:26:28.200 --> 00:26:30.200
I do not know. You just take the 2 Pi away
00:26:30.200 --> 00:26:32.200
and see where I am. It is like subtracting
00:26:32.200 --> 00:26:34.200
360 degrees.
00:26:34.200 --> 00:26:36.200
00:26:36.200 --> 00:26:38.200
Absolutely!
00:26:38.200 --> 00:26:40.200
Yes. Okay.
00:26:40.200 --> 00:26:42.200
Let us subtract another 2 Pi.
00:26:42.200 --> 00:26:44.200
00:26:45.000 --> 00:26:47.000
Which is 8 Pi over 4.
00:26:47.000 --> 00:26:49.000
Which just leaves Pi over 4.
00:26:49.000 --> 00:26:51.000
So, that means that
00:26:51.000 --> 00:26:53.000
cosine of 17 Pi over 4,
00:26:53.000 --> 00:26:55.000
is going to be the same thing
00:26:55.000 --> 00:26:57.000
as cosine of Pi over 4.
00:26:57.000 --> 00:26:59.000
00:26:59.000 --> 00:27:01.000
00:27:01.000 --> 00:27:03.000
00:27:03.000 --> 00:27:05.000
00:27:05.000 --> 00:27:07.000
Because 17 Pi over 4
00:27:07.000 --> 00:27:09.300
is really Pi over 4
00:27:09.300 --> 00:27:11.400
plus 4 Pi.
00:27:11.400 --> 00:27:13.400
00:27:13.400 --> 00:27:15.400
00:27:15.400 --> 00:27:17.400
What is the cosine of Pi over 4?
00:27:17.400 --> 00:27:19.400
00:27:19.400 --> 00:27:21.400
00:27:21.400 --> 00:27:23.400
00:27:23.400 --> 00:27:25.000
00:27:25.000 --> 00:27:27.400
How did I get the 8 Pi over 4? 2 Pi is 8 Pi over 4.
00:27:27.400 --> 00:27:29.400
00:27:29.400 --> 00:27:31.400
2 is 8 over 4.
00:27:31.400 --> 00:27:33.000
00:27:33.000 --> 00:27:35.000
It could mean 12 Pi over 6, but I want
00:27:35.000 --> 00:27:37.000
to have Pi over 4 when I am done.
00:27:37.000 --> 00:27:39.000
00:27:39.000 --> 00:27:41.000
00:27:41.000 --> 00:27:43.000
00:27:43.000 --> 00:27:45.000
00:27:45.000 --> 00:27:47.000
00:27:47.000 --> 00:27:49.000
00:27:49.000 --> 00:27:51.000
00:27:51.000 --> 00:27:53.000
00:27:53.000 --> 00:27:55.000
00:27:55.000 --> 00:27:57.000
00:27:57.000 --> 00:27:59.000
00:27:59.000 --> 00:28:01.000
How about
00:28:01.000 --> 00:28:03.000
41 Pi over 6
00:28:03.000 --> 00:28:04.000
00:28:04.000 --> 00:28:06.000
41 over 6 is bigger than 2 right?
00:28:06.000 --> 00:28:08.000
So, it is 41 Pi over 6 is
00:28:08.000 --> 00:28:10.000
bigger than 2 Pi.
00:28:10.000 --> 00:28:12.000
So, I have to figure out what angle this is.
00:28:12.000 --> 00:28:14.000
It is less than 2 Pi.
00:28:14.000 --> 00:28:16.000
Everybody agree? It gets down to
00:28:16.000 --> 00:28:18.000
some angle less than 2 Pi.
00:28:18.000 --> 00:28:20.000
So, 41 Pi over 6
00:28:20.000 --> 00:28:22.000
I will subtract 2 Pis.
00:28:22.000 --> 00:28:24.000
So, 2 Pi
00:28:24.000 --> 00:28:26.000
is 12 Pi over 6.
00:28:26.000 --> 00:28:28.000
00:28:28.000 --> 00:28:30.000
I am just doing fraction work.
00:28:30.000 --> 00:28:32.000
I am going to take 41 6ths and I am going
00:28:32.000 --> 00:28:34.000
to subtract
00:28:34.000 --> 00:28:36.000
12 Pi over 6.
00:28:36.000 --> 00:28:38.000
00:28:38.000 --> 00:28:40.000
and I get
00:28:40.000 --> 00:28:42.000
29 Pi over 6.
00:28:42.000 --> 00:28:44.000
Is 29 over 6 bigger than 2?
00:28:44.000 --> 00:28:46.000
Yes!
00:28:46.000 --> 00:28:48.000
So I need to take away 12 over 6 again.
00:28:48.000 --> 00:28:50.000
00:28:50.000 --> 00:28:52.000
00:28:52.000 --> 00:28:54.000
00:28:54.000 --> 00:28:56.000
00:28:56.000 --> 00:28:58.000
00:28:58.000 --> 00:29:00.000
Is 17 over 6 bigger than 2?
00:29:00.000 --> 00:29:03.000
Yes! I need to take away another 12 over 6.
00:29:03.000 --> 00:29:05.500
I could have done it all in one step of course.
00:29:05.500 --> 00:29:07.500
00:29:07.500 --> 00:29:09.500
00:29:09.500 --> 00:29:11.500
00:29:11.500 --> 00:29:13.500
Now I am down to 5 Pi over 6.
00:29:13.500 --> 00:29:15.500
5 Pi over 6 is
00:29:15.500 --> 00:29:17.500
not. Now I know where to stop.
00:29:17.500 --> 00:29:19.500
Cause now I am working with a number that is
00:29:19.500 --> 00:29:21.500
less than 12 Pi over 6. Less than 2 Pi.
00:29:21.500 --> 00:29:23.500
Which is 12 over 6.
00:29:23.500 --> 00:29:25.500
So, sine of 41 Pi over 6
00:29:25.500 --> 00:29:27.500
is the same as saying,
00:29:27.500 --> 00:29:29.500
sine of 5 Pi over 6.
00:29:29.500 --> 00:29:31.500
00:29:31.500 --> 00:29:33.500
So, then, we need to figure out what
00:29:33.500 --> 00:29:35.500
is the sine of 5 Pi over 6.
00:29:35.500 --> 00:29:37.500
00:29:37.500 --> 00:29:39.500
150 degrees?
00:29:39.500 --> 00:29:41.500
You do not have to convert it to degrees.
00:29:41.500 --> 00:29:43.500
But, most of you are more comfortable with that.
00:29:43.500 --> 00:29:45.500
00:29:45.500 --> 00:29:47.500
So, 150 degrees is
00:29:47.500 --> 00:29:49.500
about there.
00:29:49.500 --> 00:29:52.000
Our reference angle is 30 degrees.
00:29:53.000 --> 00:29:55.000
So, this is the same as the sine of 30 degrees
00:29:55.000 --> 00:29:57.000
or the sine of Pi over 6.
00:29:57.000 --> 00:29:59.000
00:29:59.000 --> 00:30:00.000
00:30:00.000 --> 00:30:02.000
The sine of 30 degrees is...
00:30:02.000 --> 00:30:04.000
1/2.
00:30:04.000 --> 00:30:05.500
00:30:05.500 --> 00:30:07.500
Now is it positive or negative?
00:30:08.000 --> 00:30:10.100
Positive because we are in the second quadrant.
00:30:10.100 --> 00:30:12.100
00:30:12.100 --> 00:30:14.100
So, therefore,
00:30:14.100 --> 00:30:16.100
the sine of 41 Pi over 6
00:30:16.100 --> 00:30:18.100
00:30:18.100 --> 00:30:20.100
is 1/2.
00:30:20.100 --> 00:30:22.100
00:30:22.100 --> 00:30:24.100
00:30:24.100 --> 00:30:26.100
00:30:26.100 --> 00:30:28.100
00:30:28.100 --> 00:30:30.100
00:30:30.700 --> 00:30:32.700
00:30:32.700 --> 00:30:34.700
00:30:34.700 --> 00:30:36.700
00:30:36.700 --> 00:30:38.700
00:30:38.700 --> 00:30:40.700
00:30:40.700 --> 00:30:42.700
00:30:42.700 --> 00:30:44.700
00:30:44.700 --> 00:30:46.700
00:30:46.700 --> 00:30:48.700
00:30:48.700 --> 00:30:50.700
00:30:50.700 --> 00:30:52.700
00:30:52.700 --> 00:30:54.700
00:30:54.700 --> 00:30:56.700
00:30:56.700 --> 00:30:58.700
By the way, we could always turn this into degrees
00:30:58.700 --> 00:31:00.700
and subtract 360. It is
00:31:00.700 --> 00:31:02.700
exactly the same thing.
00:31:02.700 --> 00:31:04.700
00:31:04.700 --> 00:31:06.700
You can turn it into degrees
00:31:06.700 --> 00:31:08.700
subtract 360 or you can
00:31:08.700 --> 00:31:10.700
leave it in radians and subtract 2 Pi.
00:31:10.700 --> 00:31:12.700
They are the same thing.
00:31:12.700 --> 00:31:14.700
00:31:14.700 --> 00:31:16.700
00:31:16.700 --> 00:31:18.700
We got to subtract 2 Pis.
00:31:18.700 --> 00:31:20.700
Alright, now, you can
00:31:20.700 --> 00:31:22.700
do it all at once, by the way. You could
00:31:22.700 --> 00:31:24.700
look at this and say,
00:31:24.700 --> 00:31:26.700
8 Pi over 4 is 2 Pi.
00:31:26.700 --> 00:31:28.700
So, you look at this and say, okay now you know I will need
00:31:28.700 --> 00:31:30.700
more than one 8. You may not notice that I can subtract 16 right away,
00:31:30.700 --> 00:31:32.700
but you will.
00:31:33.700 --> 00:31:35.700
So,
00:31:35.700 --> 00:31:37.700
23 Pi over 4
00:31:37.700 --> 00:31:39.700
00:31:39.700 --> 00:31:41.700
minus 2 Pi
00:31:41.700 --> 00:31:43.700
is 23 Pi over 4
00:31:43.700 --> 00:31:45.500
00:31:45.500 --> 00:31:47.700
minus 8 over 4
00:31:47.700 --> 00:31:49.700
00:31:49.700 --> 00:31:51.700
is 15 Pi over 4.
00:31:51.700 --> 00:31:53.700
00:31:53.700 --> 00:31:55.700
00:31:55.700 --> 00:31:58.700
Take away an 8 over 4 again.
00:31:58.700 --> 00:32:00.700
00:32:00.700 --> 00:32:02.700
00:32:02.700 --> 00:32:04.700
00:32:05.100 --> 00:32:07.100
Now I have 7 Pi over 4.
00:32:08.100 --> 00:32:10.100
7 Pi over 4 is
00:32:10.100 --> 00:32:12.100
less than 2 Pi. So
00:32:12.100 --> 00:32:14.100
this is the same as the cosine
00:32:14.100 --> 00:32:16.100
of 7 Pi
00:32:16.100 --> 00:32:18.100
over 4.Let's not convert this
00:32:18.100 --> 00:32:20.100
into degrees. Let us do it in radians.
00:32:21.400 --> 00:32:24.100
All the way around,
00:32:24.100 --> 00:32:26.100
is 8 fourths.
00:32:26.100 --> 00:32:28.100
So this is one fourth less than that.
00:32:28.100 --> 00:32:30.100
So, 1 Pi over 4
00:32:30.100 --> 00:32:32.100
less than
00:32:32.100 --> 00:32:34.100
M 8 Pi over 4.
00:32:34.100 --> 00:32:36.100
So, that is going to be Pi over 4.
00:32:36.100 --> 00:32:38.100
Okay?
00:32:38.100 --> 00:32:40.100
You understand that?
00:32:40.100 --> 00:32:42.100
It is 8 fourths all the way around.
00:32:42.100 --> 00:32:44.100
So, 7 fourths we have
00:32:44.100 --> 00:32:46.100
1 fourth left over.
00:32:46.100 --> 00:32:48.100
So, this is the same as
00:32:48.100 --> 00:32:50.100
the cosine of Pi over 4
00:32:50.100 --> 00:32:52.100
which is
00:32:52.100 --> 00:32:54.100
00:32:54.100 --> 00:32:56.100
00:32:56.100 --> 00:32:58.100
00:32:59.100 --> 00:33:01.100
Why is it not 3 Pi over 4?
00:33:01.100 --> 00:33:03.100
Because, once I subtracted
00:33:03.100 --> 00:33:05.300
the 2 Pis I got down to 7 Pi over 4
00:33:05.800 --> 00:33:07.800
Okay, and that is where I stopped because that number
00:33:07.800 --> 00:33:09.800
is less than 2 Pi.
00:33:09.800 --> 00:33:11.800
00:33:11.800 --> 00:33:13.800
Well, it is okay to say
00:33:13.800 --> 00:33:15.800
negative Pi over 4.
00:33:15.800 --> 00:33:17.800
00:33:17.800 --> 00:33:19.800
Yes.
00:33:19.800 --> 00:33:21.800
Is it positive or negative?
00:33:21.800 --> 00:33:23.800
Good.
00:33:23.800 --> 00:33:25.800
Okay.
00:33:25.800 --> 00:33:27.800
00:33:27.800 --> 00:33:29.800
00:33:29.800 --> 00:33:31.800
So, where did I get 8 fourths from? I am doing
00:33:31.800 --> 00:33:33.800
fractions. Okay? That is 23 fourths.
00:33:33.800 --> 00:33:35.800
And I need to take away 2.
00:33:35.800 --> 00:33:37.800
So, 23 fourths minus 2
00:33:37.800 --> 00:33:39.800
is 23 fourths minus
00:33:39.800 --> 00:33:41.800
8 fourths is 15 fourths.
00:33:41.800 --> 00:33:43.800
Okay?
00:33:44.500 --> 00:33:46.500
15 fourths minus 2.
00:33:46.500 --> 00:33:48.500
Is 15 fourths minus 8 fourths.
00:33:48.500 --> 00:33:50.500
7 fourths.
00:33:50.500 --> 00:33:52.500
00:33:52.500 --> 00:33:54.500
So, I covered almost everything
00:33:54.500 --> 00:33:56.500
that I wanted to cover today.
00:33:56.500 --> 00:33:58.500
But, not everything.
00:33:58.500 --> 00:34:00.500
So, Wednesday we are going to do graphs.