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Course: Precalculus>Unit 2

Lesson 2: Trigonometric identities on the unit circle

Sine & cosine identities: periodicity

Sal finds trigonometric identities for sine and cosine by considering angle rotations on the unit circle. Created by Sal Khan.

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• What are some other relationships besides the one that Sal came up with at the end of the video?
• Here are some trigonometric relationships that can be found by playing around with the unit circle:
sin(pi-θ)=sin(θ)
cos(pi-θ)= -cos(θ)
sin(θ+pi)= -sin(θ)
cos(θ+pi)= -cos(θ)
cos(θ+pi/2)= -sin(θ)
sin(-θ)= -sin(θ)
cos(-θ)=cos(θ)

• Is sin always related to the y axis? I'm confused because if we look at angle theta + pi/2, why is it that sin doesn't have the opposite/hypotenuse definition? If sin is opposite side/hypotenuse wouldn't the sin of theta + pi/2= sin of theta? The opposite side of the "triangles" is the same isn't it?
• Yes, sine is directly related to the y axis. When an angle intersects the unit circle, the sin is equal to the y value of the point at which it intersects.
Sine (theta+pi/2) is equal to cosine.
• Is it me, or did Sal mistake his notation in the final part of the video in concluding that cos(theta) = sin(theta+pi/2)? Because that is not a right triangle, and would need to be analyzing its complimentary angle (90-theta) since 90-theta+theta+pi/2=180 degrees. Since sin(90-theta) = cos(theta), and cos(90-theta) = sin(theta) this seems to be the right way... anyone else notice this or am I mistaken?
• Sine, cosine and the other functions are not just defined for right angles, though the simple definitions you start with for these functions only work for the acute angles of right triangles.

But, yes, cos x = sin(x + ½π)
It is also true that cos x = sin(½π - x)
Thus, it is true that
sin(½π - x) = sin(x + ½π)

It is also true that cos(½π-x) = sin(x)
and that cos(x- ½π) = sin(x)
So, cos(½π-x) = cos(x- ½π)

Sine and cosine are both periodic functions that are identical except for being shifted ½π radians out of phase. Thus, there are a number of ways you can shift them around to be in phase and therefore equal.

Note: Do not mix degrees and radians. Once you can use radians, you should just drop degrees altogether because in math that isn't too far down the road, degrees just won't work, you have to use radians.
• At Sal says that the length of the magenta line is Sine theta plus Pi. How did he come to that?

Wouldn't the length of that line be equal to Cos theta since we are rotating the triangle?
• Yes, you're saying the same thing Sal said - that they're equal.

By using the unit circle definition of sin(a) and cos(a) ( = y/r and x/r at angle a);
sin(t+pi/2) = cost
• I think there is a good way for us to remember these trig relationships by a "Chinese Magical term 奇变偶不变，符号看象限”, which means that:
1. the change between sin and cos is based on the angle (x + θ) (in this case, if the number "x" is the 90 degree's odd multiple, such as 270 degree that is 3 times of 90 degree, the sin will be changed into cos while the cos will be changed into sin.
For example: sin(θ) = cos(270 + θ) because "270 = 90 x 3, 3 is odd"
cos(θ) = sin (450 + θ) because " 450 = 90 x 5, 5 is odd"

2. On the other hand, if the y, in the angle (y + θ), is the even multiple to 90 degree, such as pi, 180 degree, 360 degree, etc., the sin or cos will not be changed.
For example: sin(θ) = sin (180 + θ), because "180 = 90 x 2, 2 is even"
cos(θ) = cos (540 + θ), because "540 = 90 x 6, 6 is even"

3. the positive or negative is based on the initial degree's sign, such as angle (180 + θ), in this case, if θ is an angle in the first quadrant, which means that both of sin(θ) and cos(θ) are positive, the sin(180 + θ) will be sin(θ) without being changed, because " θ is in the first quadrant and 180 = 2 x 90, 2 is even, according to the previous sentence.
For example:
sin120 = sin (90 + 30) = cos (90 x 1 + 30), because 1 is odd, and sin120 is in the second quadrant, corresponding to positive sign.
sin600 = sin(6 x 90 + 60) = sin60, because 6 is even, and sin60 is in the first quadrant, corresponding to the positive sign.
-cos150 = -cos(90 x 1 + 60) = -sin60, because 1 is odd, and -cos150 is in the third quadrant, corresponding to the negative sign.

At the end, thanks for the Khan Academy's service about SAT and other tremendously meaningful programs!
• Why do people name angles with Greek letters?
• Probably because many geometry concepts were figured out by Greek mathematicians. Euclid (a Greek mathematician) is often referred to as the father of geometry.
• does cos(θ+π/2) = sin (θ) ?
• Not quite. Remember the direction in which you want the function to move. Since you want to move the cosine function forward, you have to subtract the pi/2 from the theta value.

So, sin(theta) = cos(theta - pi/2).

What you have there will be -sin(theta)
• At there's a dotted horizontal line above the second quadrant which would have supposedly made a triangle. How come sin wasn't on that dotted line and instead was on the y-axis? Shouldn't it be on that dotted line?
• That triangle is not related to the sin that we are looking for. We are looking for the sin(θ+π/2), and we can find it by looking at the y-coordinate of the point on the ray which intersects the circle. This y-coordinate is also the vertical distance from said point to the origin. This is exactly the line that was highlighted in pink.

The horizontal line you mentioned is actually sin (θ). Imagine that triangle you pointed out being rotated clockwise by 90° at the origin. The two triangles map onto one another.
• Sal says that the unit circle definition is an extension of soh-cah-toa, but wouldn't it be the other way around? Isn't soh-cah-toa an extension of the unit circle definition?
(1 vote)
• I guess you could think of it both ways. Both definitions are derivable from the other definition. For example, you can say:
Sin(x) = Opposite/Hypotenuse
(In a unit circle, Hypotenuse=1, so)

> Sin(x) = Opposite/1
> Sin(x) = Opposite
In this case, the length of the side opposite the central angle = the y length of the triangle.
>> that means that the y coordinate (Where the Hypotenuse intersects the circle) = Sin(x)
>>>the same logic applies for cos(x)

The reason some people say that the unit circle definition is an extension of Soh-Cah-Toa is mostly because Soh-Cah-Toa (I believe) was defined first, and the Unit circle definition is applicable to more scenarios(Any triangle versus only right triangles). The Unit circle definition EXPANDS upon the Soh-Cah-Toa definition. It takes you more places.
• In the video, Sal proved that:

`sin(π/2 + θ) = cos θ`

But, in my textbook, the identity given goes like this;

`sin(π/2 - θ) = cos θ`

So does that mean that `sin(π/2 - θ) = sin(π/2 + θ)` or is there a mistake somewhere?