WEBVTT
Kind: captions
Language: en
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Trigonometry, do you know when you will use trigonometry?
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NEVER. (laughter)
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You will never use it. It was invented to destroy the hopes and dreams of young people. (laughter)
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So, its actually useful in things like physics. So, when you take your physics class or when you become a doctor you will need trigonometry.
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I don't think it shows up in chemistry, but definitely for those of you who plan on becoming engineers you will need to know trigonometry.
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So what is this trig stuff? Most of you have seen this already. You may remember the unit circle.
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So, lets do it again. So first we do what is called right triangle trigonometry.
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The idea is the following:
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Thousands of years ago and you are ancient Babylonians and you are making war on the Hittites or the Philistines or whoever, and you want to knock down their wall...
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Because they have a big wall around their city, and you could knock it down and get in there and get all of their loot. Right?
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So, the goal, however, is to not get shot by the flaming arrows.
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You want to come up with a way to knock down the wall. The trick is...
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How can you figure out how high the wall is and how far away you are?
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When trigonometry was invented people realized that it was sort of a ratio and it stayed the same no matter where you were.
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You remember in geometry, there was a rule of similar triangles.
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The rule was that the sides are proportional.
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Say you have that little picture right there. Let's just call this angle, 20 degrees.
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It doesn't matter what it equals.
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So you are standing here.
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The wall is over there and you want to figure out how high that wall is.
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So people measured this distance.
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People measured that distance a long time ago without getting the flaming arrows.
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They stood back and knew that if you shot an arrow, then you would know how far that arrow was going. because they were good at that.
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So you figure out how far away you are from something.
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And, you figure out how tall this was.
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Then they realized because it is similar triangles.
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That this ratio was the same as this ratio.
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In other words, y/x is the same as B/A
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In fact, you can even create a little triangle. The ratio is the same as C/D.
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And so on, they discovered that all of these ratios are the same.
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So, once you had an angle. This ratio never changed as long as you were dealing with similar triangles.
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Okay?
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By the way, how do you know that triangles are similar? We learned this in geometry.
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Triangles are similar, if among other things, if all the angles are equal.
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This triangle here has a 20 degree angle, and a 90 degree angle.
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That means the other one has to be 70 degrees.
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And, this one has a 20 degree and 90 degree angles.
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So, that has to be 70.
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So, those triangles have to be similar.
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And, once they are similar, you know that the sides always have the same ratios.
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So, they gave names to these ratios.
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The ratios they named, they named them sine, cosine, and tangent.
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So, that is what we are going to learn.
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The first most important ratio to learn is called sine.
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So, if you have any triangle and you have an angle, which we will call x,
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Then the sine of x, written: sin(x), is the ratio of this side (A) to that side (C).
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The cosine of x, written: cos(x), is the ratio of B/C.
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There is an angle mnemonic to remember these things.
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Tangent of x, written: tan(x), is A/B.
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You see the muscles there, that is called the gun show. (giggles)
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Those are the three ratios that you will need.
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Whats's our mnemonic?
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I bet most of you have learned SOH CAH TOA.
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They taught you in school. It goes like this.
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What does it stand for?
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This says that sine of any angle is the length of the side opposite divided by the length of the hypotenuse.
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This needs to be sine of an angle because you, technically, cannot just have sine. It needs to be sine of x.
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The cosine of the angle is the length of the adjacent side divided by the length of the hypotenuse.
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The tangent of the angle is the length of the opposite side divided by the length of the adjacent side.
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What do we mean by opposite and what do we mean by adjacent?
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Say you are standing there and you are measuring up the wall. That wall is opposite you.
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When I look at the wall, it is opposite me.
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When you think of this angle, this is the opposite side.
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The hypotenuse is the side opposite the 90 degree angle.
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This is the hypotenuse.
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The adjacent is the other side.
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It is adjacent to me. I can touch it.
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But, basically it is the leftover.
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The third side is the adjacent.
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The sine of any angle is the opposite divided by the hypotenuse.
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The cosine is the adjacent side over the hypotenuse.
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The tangent is the opposite side over the adjacent side.
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For example,
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Lets say I have some random triangle and this is the angle x.
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This side is length 6.
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This side is length 8.
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First of all, if I know those two sides, I can learn how to find the hypotenuse. Right?
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There is two formulas that you memorized in class.
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You memorized the quadratic formula. Remember that one?
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x= -b + or - the square root of b squared minus 4ac all over 2a.
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There is a little song.
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Then, there is the Pythagorean Theorem.
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You guys don't know any others, right?
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Maybe you do, but probably not.
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But, I bet you all know the Pythagorean Theorem.
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Right? Which is a^2 +b^2=c^2, of course, depending on where a, b and c are.
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Ok, so...
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We only need to be given 2 sides of a triangle.
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You can, th en, always figure out the third.
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So, if this is 6 and this is 8. Then, you can find that side.
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That's very impressive.
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6 squared plus 8 squared is c squared.
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So, 36 plus 64...
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100 is c squared.
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So, c is 10.
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c could also technically be negative 10,
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but, in geometry we don't use negative numbers.
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They don't make much sense.
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So, we stick with positive 10.
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Ok? But, now that we know that this is 10,
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We can find the sine and cosine and tangent of x.
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Sine of x is the opposite side, 6,
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Remember, you look at the angle and look at the side opposite it and get 6.
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Divided by the hypotenuse.
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Cosine of x is
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the adjacent over the hypotenuse of an angle.
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Sure.
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Okay. Sure.
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Remember that.
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8 over 10
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Its the other side over 10.
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And, then, the tangent
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is the opposite over the adjacent.
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You do not need to simplify these.
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If this were a test and you wrote 6 over 8 that's fine.
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You do not need to reduce it to 3 over 4.
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Okay?
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You can if you want. You have no obligation to reduce.
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Webassign will reduce for you automatically.
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Okay, so if you have something horrible,
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on Webassign, 196 over 300,
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You don't need to reduce that. Okay?
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This reduces to
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49 over 75.
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Okay? But, you don't actually need to reduce it. You can just put it in that form.
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Be careful with decimals in Webassign.
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Because, you know, you might not be exactly right.
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People have a tendency to round their decimals too much.
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Oh! That reminds me.
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You can use a calculator for any part of this class, but not in exams.
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Um, anyway. So,
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you do not need to reduce things.
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These are numbers. We are going to try to keep the numbers nice and simple and not messy. Okay?
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Um, in fact if you are doing a problem
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and getting really bad numbers, you are probably doing it wrong.
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Okay?
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These should come out as very straight forward numbers.
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If you are not sure,
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just kind of circle it and say this is kind of what I am doing.
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and you will receive partial credit.
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And, at least you will get that part correct.
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Okay, let's have you guys do another one of these.
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Ah!
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See that. That's the guns. (laughter)
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Um, see you sit in the front and you get extra attention.
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Its really good. You got to remember not to do that again.
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Don't make that mistake twice.
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Let's say that we give you x, and
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this is, ah, 12
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and that is 13.
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You have to find the sine, cosine, and tangent.
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Alright, so, to find the sine, cosine, and tangent.
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First thing you do, you need to figure out the missing side.
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So, you do the Pythagorean Theorem.
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So, you say, A squared plus 12 squared is 13 squared.
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Use your calculator.
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Okay? So, if you solve this,
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You get A equals
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5
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Okay? so this is a 5, 12, 13 triangle.
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So far, so good?
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Alright, now to find the sine...
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You do opposite over hypotenuse
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So, you do 12 divided by 13.
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Did you get that one?
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Yay! Because you did this in high school, right?
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Okay. Cosine...
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You do the adjacent over the 13.
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Then, tangent. You do the opposite over the adjacent.
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Simple?
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What about using this angle, y?
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Now, find the sine, cosine, and tangent.
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The sine of y is
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would be 5 over 13.
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Notice
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its the same thing as the cosine of x.
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The sine of one angle is the cosine of the other angle.
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That's what the co actually stands for.
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Co means complementary.
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Remember, from geometry, complementary angles are angles that add to 90 degrees.
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So, these 2 angles have to add to 90 degrees.
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The angles in triangles add up to 180.
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And, that's 90.
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So, one angle is the complement to the other.
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So, the sine of one is the cosine of the other.
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Okay? The cosine of y is 12 over 13.
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Because if you look at y and say adjacent
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over hypotenuse.
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Tangent of y
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is 5 over 12.
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So far, so good?
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Ah. Will we be ever asked to do sine, cosine, and tangent of the right angle?
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No. Okay.
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Because it doesn't really make sense.
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Okay.
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Remember when I told you that the sine of x is
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opposite over the hypotenuse.
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Sine of x is
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A over C.
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Cosine of x
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is B over C.
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There is a relationship there in the Pythagorean Theorem that is coming. Okay?
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What if I put these on a circle instead?
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I put them on
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Something like that.
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Okay? And I said,
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Let's just pick some random coordinate.
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I guess I should just call them x and y.
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Alright, so that's
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a coordinate x and y.
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The coordinates of this point
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are (x,y).
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If I make that just 1,
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What do I know about the sine of this angle down here?
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Which we use the greek letter theta.
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Do you know why we use theta?
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I don't know. We just use greek letters.
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I guess we did not want to use the other letters.
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Okay?
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We use the greek letter theta
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to stand for the angle that is made by
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the radius and the x-axis.
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The sine of that angle
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is y divided by 1.
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Do you guys see that?
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Let's blow it up.
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Make it a little bigger.
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The radius of this is 1.
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And, if you pick any point on this circle,
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this circle is called the unit circle.
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Unit just means 1.
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Okay?
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The coordinates are x and y.
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Because the coordinates of any point are x and y.
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Think about the opposite for a minute.
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You say the sine of theta
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is the opposite side which is y
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divided by the hypotenuse.
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The hypotenuse is just 1.
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So, the sine of theta is just going to be y.
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The cosine of theta is just x.
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Okay?
00:15:36.000 --> 00:15:38.000
So in other words,
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if I know the coordinates of a point,
00:15:40.000 --> 00:15:42.000
on that circle, then you know
00:15:42.000 --> 00:15:44.000
the sine and cosine of that angle.
00:15:44.000 --> 00:15:46.000
This will be very handy
00:15:46.000 --> 00:15:48.000
if I want to find sine and cosine for angles
00:15:48.000 --> 00:15:50.000
that don't sit inside of a right triangle.
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The problem with a right triangle is that it only goes up to 90 degrees.
00:15:54.000 --> 00:15:56.000
So, how do you find sine and cosine
00:15:56.000 --> 00:15:58.000
bigger than 90 degrees?
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You do this.
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What else do we notice about the unit circle?
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I going to have to pull that back down.
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If you have a point in the first quadrant,
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what do you know about the coordinates of those points?
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They are both positive.
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Okay? So,
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the x-coordinate and y-coordinate are both positive in the first quadrant.
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In the second quadrant, over here.
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The x-coordinate is now negative.
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Because we are going to the left. The y-coordinate is still positive.
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Okay, so
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the coordinates would be
00:16:39.000 --> 00:16:41.000
a negative value and a positive value.
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Down here the coordinates
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will be a negative value and a negative value.
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And, down here
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positive and negative.
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So, remember the x-coordinate
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is your cosine
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and the y-coordinate is your sine.
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So, this tells you
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That in the first quadrant,
00:17:03.000 --> 00:17:05.000
sine and cosine are both positive.
00:17:14.000 --> 00:17:16.000
In the second quadrant,
00:17:16.000 --> 00:17:18.000
the sine
00:17:18.000 --> 00:17:20.000
will stay positive
00:17:20.000 --> 00:17:22.000
but the cosine will be negative.
00:17:22.000 --> 00:17:24.000
Why will it be negative?
00:17:24.000 --> 00:17:26.000
Because you will be going this way.
00:17:26.000 --> 00:17:28.000
And, x-coordinates will be negative over here.
00:17:28.000 --> 00:17:30.000
And, the x-coordinate is cosine.
00:17:30.000 --> 00:17:32.000
We will be doing a lot more of this on Wednesday.
00:17:32.000 --> 00:17:34.000
So, don't be scared.
00:17:34.000 --> 00:17:36.000
In the third quadrant,
00:17:36.000 --> 00:17:38.000
they will both be negative.
00:17:44.000 --> 00:17:46.000
In the fourth quadrant,
00:17:46.500 --> 00:17:49.000
the sine is negative
00:17:51.000 --> 00:17:53.000
and the cosine is positive.
00:17:53.000 --> 00:17:55.000
Okay?
00:17:56.000 --> 00:17:58.000
So, if I tell you what quadrant
00:17:58.000 --> 00:18:00.000
a point is in
00:18:00.000 --> 00:18:02.500
which we will figure out more of this on Wednesday.
00:18:02.500 --> 00:18:05.000
Then, you can tell if it is positive or negative.
00:18:05.000 --> 00:18:07.000
So, for the moment we are going to stick
00:18:07.000 --> 00:18:09.000
with the first quadrant, where everything is positive.
00:18:13.000 --> 00:18:17.000
I am going to teach you one other right triangle trigonometry for today.
00:18:17.000 --> 00:18:20.000
That's what I am going to do for the rest of the day.
00:18:29.000 --> 00:18:33.000
One of our favorite triangles is the 30, 60, 90 triangle.
00:18:33.000 --> 00:18:36.000
Do you know why we use the 30, 60, 90 triangle?
00:18:36.000 --> 00:18:38.000
Well,
00:18:38.000 --> 00:18:40.000
If we have an equilateral triangle,
00:18:46.000 --> 00:18:50.500
You know that's 60 degrees, that's 60 degrees, and that's 60 degrees.
00:18:50.500 --> 00:18:52.500
Right? An equilateral triangle.
00:18:52.500 --> 00:18:55.000
So, you cut it in half like this.
00:18:57.000 --> 00:18:59.000
You get a,
00:18:59.000 --> 00:19:01.000
you cut the triangle directly in half
00:19:01.000 --> 00:19:03.000
perpendicular to the base
00:19:03.000 --> 00:19:05.000
you get a right angle.
00:19:05.000 --> 00:19:07.000
This angle becomes 30 degrees and that is 30 degrees.
00:19:07.000 --> 00:19:10.000
And, you get a 30, 60, 90 triangle.
00:19:10.000 --> 00:19:12.500
Okay, 30 degrees, 60 degrees, 90 degrees.
00:19:12.500 --> 00:19:14.500
Furthermore,
00:19:14.500 --> 00:19:17.000
This is an equilateral triangle,
00:19:17.000 --> 00:19:19.000
so this distance
00:19:19.000 --> 00:19:21.000
is half of this distance.
00:19:21.000 --> 00:19:23.000
So, if this is x
00:19:24.500 --> 00:19:27.000
The hypotenuse is 2x.
00:19:27.000 --> 00:19:29.000
Okay?
00:19:29.000 --> 00:19:31.000
In an equilateral triangle, that is 2x
00:19:32.000 --> 00:19:34.000
and that is 2x.
00:19:34.000 --> 00:19:37.000
And, if you use the Pythagorean Theorem.
00:19:37.000 --> 00:19:39.000
This side
00:19:39.000 --> 00:19:42.000
is x times the square root of 3.
00:19:42.000 --> 00:19:44.000
That is a 3.
00:19:46.000 --> 00:19:48.000
So, that is a 30, 60, 90 triangle.
00:19:48.000 --> 00:19:51.000
And, those ratios never change.
00:19:51.000 --> 00:19:53.000
So,
00:19:53.000 --> 00:19:55.500
If this is a 30, 60, 90 triangle
00:19:55.500 --> 00:19:58.000
and this is x that is going to be 2x.
00:19:58.000 --> 00:20:00.500
and that is going to be x times the square root of 3.
00:20:00.500 --> 00:20:02.500
That comes from the Pythagorean Theorem
00:20:02.500 --> 00:20:05.500
and the fact that we are cutting an equilateral triangle in half.
00:20:06.000 --> 00:20:09.500
So, let's find the sine, cosine, and tangent of all of this.
00:20:23.000 --> 00:20:25.000
Okay. Sine of x
00:20:25.000 --> 00:20:28.000
will be opposite divided by the hypotenuse.
00:20:29.500 --> 00:20:32.000
So, x over 2x.
00:20:32.000 --> 00:20:34.000
Which will be 1/2.
00:20:36.000 --> 00:20:38.000
Cosine of x
00:20:38.000 --> 00:20:41.000
is the x square root of 3 over 2x.
00:20:45.000 --> 00:20:47.500
Which works out to the square root of 3 over 2.
00:20:51.000 --> 00:20:53.500
Tangent of x
00:20:53.500 --> 00:20:57.500
will be x over x times the square root of 3.
00:20:59.000 --> 00:21:02.500
Which will reduce to 1 over the square root of 3.
00:21:04.000 --> 00:21:10.000
I don't know why I wrote x. These should be sine, cosine, and tangent of 30 degrees.
00:21:10.500 --> 00:21:12.500
That is why you don't do things in ink.
00:21:14.500 --> 00:21:16.500
So, you use pencil.
00:21:16.500 --> 00:21:18.500
Okay?
00:21:18.500 --> 00:21:23.500
So, these are very handy to memorize.
00:21:25.500 --> 00:21:29.500
Part of what we will do on Wednesday is to come up with a good way to memorize this.
00:21:29.500 --> 00:21:31.500
Its easy. We will have to draw a triangle.
00:21:33.500 --> 00:21:36.500
The sine of 30 degrees will always be 1/2.
00:21:38.000 --> 00:21:40.500
The cosine of 30 degrees
00:21:40.500 --> 00:21:43.000
will always be the square root of 3 over 2.
00:21:43.000 --> 00:21:45.500
And, tangent of 30 degrees.
00:21:46.500 --> 00:21:49.000
will always be 1 over the square root of 3.
00:21:53.000 --> 00:21:57.000
Remember what I told you before about the sine and the cosine.
00:21:57.000 --> 00:21:59.000
The sine of this angle
00:21:59.000 --> 00:22:01.000
is the cosine of that angle.
00:22:01.000 --> 00:22:03.000
So, the sine of 30 degrees
00:22:03.000 --> 00:22:05.000
is the cosine of 60.
00:22:05.000 --> 00:22:07.500
So, the sine of 60 degrees.
00:22:14.000 --> 00:22:16.500
is the x root 3 over 2x.
00:22:16.500 --> 00:22:18.500
So that is going to become square root of 3 over 2.
00:22:23.000 --> 00:22:25.000
The cosine of 60 degrees
00:22:25.000 --> 00:22:27.500
is 1/2. See how those are switching places?
00:22:29.500 --> 00:22:31.500
And, the tangent of 60 degrees
00:22:34.500 --> 00:22:37.000
Well, let's see. Tangent
00:22:37.000 --> 00:22:39.000
is the opposite over the adjacent.
00:22:39.000 --> 00:22:41.000
which is
00:22:41.000 --> 00:22:44.000
x root 3 over x.
00:22:47.500 --> 00:22:49.500
Which is just root 3.
00:22:49.500 --> 00:22:51.500
Okay? So...
00:22:51.500 --> 00:22:53.500
these are things you should memorize.
00:22:55.500 --> 00:22:57.500
So far, so good?
00:22:57.500 --> 00:22:59.500
Questions?
00:23:07.500 --> 00:23:09.500
One last thing.
00:23:19.500 --> 00:23:21.500
Okay. We can have
00:23:21.500 --> 00:23:23.500
this 45, 45, 90 triangle.
00:23:23.500 --> 00:23:25.500
That comes out of a square.
00:23:27.500 --> 00:23:30.000
In a square you have a diagonal.
00:23:30.000 --> 00:23:32.500
These 2 sides are the same.
00:23:34.000 --> 00:23:36.000
And you do the Pythagorean Theorem,
00:23:36.000 --> 00:23:38.000
That comes out x root 2.
00:23:42.000 --> 00:23:44.000
Now, the sine of 45
00:23:44.000 --> 00:23:52.000
will be x over x radical 2.
00:23:52.000 --> 00:23:54.000
Also, known as
00:23:54.000 --> 00:23:56.000
1 over radical 2.
00:24:00.000 --> 00:24:02.000
And, if you want to show off,
00:24:04.000 --> 00:24:06.000
you can rationalize that to be
00:24:06.000 --> 00:24:08.000
radical 2 over 2.
00:24:08.500 --> 00:24:10.500
Cosine of 45
00:24:14.000 --> 00:24:16.500
will also be x over x radical 2.
00:24:16.500 --> 00:24:18.500
So it will also be
00:24:18.500 --> 00:24:20.500
1 over radical 2
00:24:20.500 --> 00:24:22.500
or radical 2 over 2.
00:24:23.500 --> 00:24:26.000
Tangent of 45
00:24:30.000 --> 00:24:32.000
well, tangent is opposite over adjacent.
00:24:32.000 --> 00:24:34.000
Notice the sides are the same.
00:24:34.000 --> 00:24:36.000
it will be 1.
00:24:43.000 --> 00:24:45.000
So, this class is not review for everybody.
00:24:45.000 --> 00:24:46.500
Just for some.
00:24:48.000 --> 00:24:50.500
What you want to do...
00:24:52.500 --> 00:24:54.500
And, I really recommend you do this.
00:24:54.500 --> 00:24:56.500
So, when the exam rolls around.
00:24:56.500 --> 00:24:58.500
Because, remember, you will be nervous.
00:24:58.500 --> 00:25:00.500
You will want to have a way to instantly
00:25:00.500 --> 00:25:02.500
know the sines, cosines, and tangents.
00:25:02.500 --> 00:25:04.500
You make a little grid.
00:25:18.500 --> 00:25:21.500
30, 45, 60 across the top.
00:25:21.500 --> 00:25:24.500
Sine, cosine, and tangent down the side.
00:25:24.500 --> 00:25:26.500
Okay.
00:25:32.000 --> 00:25:34.500
Sine and cosine all have 2 as their denominators.
00:25:36.000 --> 00:25:38.500
Sine goes 1, 2, 3.
00:25:38.500 --> 00:25:40.500
1, 2, 3
00:25:40.500 --> 00:25:42.500
Great.
00:25:42.500 --> 00:25:44.500
Cosine
00:25:44.500 --> 00:25:46.500
goes 3, 2, 1
00:25:48.000 --> 00:25:50.500
Just like that. Did anyone learn that in school?
00:25:51.000 --> 00:25:53.000
Yeah, some of you. Okay.
00:25:53.000 --> 00:25:55.000
Okay, good.
00:25:55.000 --> 00:25:57.000
Not enough of you.
00:25:57.000 --> 00:25:59.000
This is the way that it was taught to me in school.
00:25:59.000 --> 00:26:01.000
Because you guys learn the unit circle
00:26:01.000 --> 00:26:03.000
you kind of stand there and have to draw it in your head.
00:26:03.000 --> 00:26:06.000
And you have no idea what you are doing.
00:26:06.000 --> 00:26:08.000
But, if you can memorize this,
00:26:08.000 --> 00:26:10.000
you can write this down for the exam.
00:26:10.000 --> 00:26:12.000
Put it on the corner of your paper and you will be fine.
00:26:12.000 --> 00:26:14.000
Tangent
00:26:14.000 --> 00:26:18.000
is found by taking sine, and dividing it by the cosine.
00:26:18.000 --> 00:26:20.000
So, the denominators will cancel.
00:26:20.000 --> 00:26:22.000
So, this will come out 1 over the square root of 3.
00:26:23.500 --> 00:26:26.000
square root of 2 over 2 over square root of 2 over 2 will come out 1.
00:26:26.000 --> 00:26:28.000
Square root of 3 over 1
00:26:29.000 --> 00:26:30.500
is the square root of 3.
00:26:34.000 --> 00:26:39.000
Tangent, the tangent is found by taking the sine and dividing it by cosine.
00:26:39.000 --> 00:26:42.000
You take the numerator of this and divided it by the numerator of this.
00:26:42.000 --> 00:26:44.000
You get 1 over the square root of 3.
00:26:44.000 --> 00:26:45.000
Okay?
00:26:45.000 --> 00:26:48.000
And, on Wednesday, I will show you why tangent is sine over cosine.
00:26:48.500 --> 00:26:52.000
You can kind of group all this in one.
00:26:54.000 --> 00:26:56.000
Questions?
00:26:56.000 --> 00:26:59.000
Guys, just feel free to ask questions.
00:26:59.000 --> 00:27:01.000
No?
00:27:04.000 --> 00:27:07.500
Well, we will get to more of these. So, the question is "what if it is not one of these angles".
00:27:07.500 --> 00:27:09.500
So, for the moment, we will just deal with these,
00:27:09.500 --> 00:27:11.500
because no calculator and we
00:27:11.500 --> 00:27:13.500
are just starting out.
00:27:13.500 --> 00:27:16.500
So, on that note it is enough for the first day.