Question 1

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?

Accepted Answer

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.

Question 2

What are some other relationships besides the one that Sal came up with at the end of the video?

Accepted Answer

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)

I hope these relationships help you in trig!

Question 3

At 4:07 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?

Accepted Answer

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

Question 4

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?

Accepted Answer

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.

Question 5

Why do people name angles with Greek letters?

Accepted Answer

Probably because many geometry concepts were figured out by Greek mathematicians.  Euclid (a Greek mathematician) is often referred to as the father of geometry.

Question 6

At 4:08 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?

Accepted Answer

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.

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.

Question 7

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?

Accepted Answer

Yes, it does mean that.
sin(π/2 - θ) = sin(π/2 + θ) can be interpreted as saying that the sine function has mirror symmetry about the line x = π/2, which I think you'll agree is the case.

Question 8

does cos(θ+π/2) = sin (θ) ?

Accepted Answer

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)

Question 9

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?

Accepted Answer

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.

Precalculus

Course: Precalculus > Unit 2

Sine & cosine identities: periodicity

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