Question 1

Do you have a full picture of a unit circle... please help with this :)

Accepted Answer

https://www.khanacademy.org/computer-programming/radians-explanation/5603450489733120

`This should help!`

Question 2

I understand how to apply the trig ratios in their purest form (absolute and a standard right angle triangle).

I also understand the theory of using the unit circle to extend the domain of sin(theta) and cosine(theta) to all real values.

With that said, how is a calculator able to take this basic principle and compute the logic that cos(89) = 0.017 and cos(91) = -0.017 (a reflection across the y-axis). What's going on under the hood?

Accepted Answer

Hi kimberm1,

I would assume that they created an algorithm that checks the angle just like a human would.
Aka first calculate the absolute value of cos(theta), and then check to see what the value of theta is.

Here is some PSUEDO CODE that might help you understand.

IF (theta >= 0 AND theta < 90) THEN
    ANSWER = cos(theta) // leave unchanged
IF (theta >= 90 AND theta < 180) THEN
    ANSWER = cos(theta) * -1

And so on. What the computer would essentially do is follow an algorithm that checks what quadrant the point will be in based upon the angle. We can easily calculate that as seen above.
Knowing the quadrant, we know know what sign the y and x values are.

Hope this helps,
- Convenient Colleague

Question 3

In the appendix, can someone explain to me WHY cotangent, tangent, secant, and cosecant are represented in that way?

Accepted Answer

tangent=sin(θ)/cos(θ)(tangent is the path from point(x,y) to the secant)
cotangent=1/tan(θ)(cot α 1/tan)(cotangent is the path from the point (x,y) to the cosecant)
secant=1/cos(θ)(sec α 1/cos)
cosecant=1/sin(θ)(csc α 1/sin)

Question 4

Why and how are trigonometric ratios related to circles?

Accepted Answer

That's the subject of this video: https://www.khanacademy.org/math/algebra2/trig-functions/unit-circle-definition-of-trig-functions-alg2/v/unit-circle-definition-of-trig-functions-1, particularly around 2:25.

The cool thing about a unit circle (a circle with radius one), is that if you draw a right triangle where the hypotenuse of the triangle connects a point on the unit circle with the point (0, 0), the length of the hypotenuse is always going to be one. Because that's the radius of the circle! And the length of the other two sides are just going to be the x and y coordinates of the point on the unit circle.

Take a little time drawing right triangles were the hypotenuse connects the middle of the unit circle to a point on the circle and see if you can convince yourself that this is the case.

So you have a right triangle where one side length is the x-coordinate and one side length is the y-coordinate, and the hypotenuse is equal to 1. This means you can figure out the sine and cosine of the angle that the hypotenuse and the x-axis make:

The cosine is the length of the adjacent side (which is the x-coordinate) divided by the length of the hypotenuse (which is 1). So the cosine is just the x-coordinate!

Similarly, the sine is the length of the opposite side (which is the length of the y-coordinate) divided by the length of the hypotenuse (which is 1). So the sine is the y-coordinate.

The trigonometric ratio defining the tangent (opposite over adjacent) may be less intuitive from the unit circle, but I hope you can see from the unit circle why the tangent of an angle is equal to the sine of the angle over the cosine.

Hope this helps.

Question 5

Why is it that you don't use the actual angles when finding tangent

Accepted Answer

It's just easier to think of it as sin/cos. After all, the unit circle only defines cosine and sine.

Question 6

So if sine, cosine, and tangent are SOH CAH TOA, could we say that secant, cosecant, and cotangent are SHO CHA TA0?

Accepted Answer

Almost. Secant is the reciprocal function of cosine, and cosecant that of sine, so it would actually read CHO SHA CAO. Probably not best to memorize them like this, though, because you already have one trig mnemonic and they mightget jumbled up. Plus, cosecant and cotangent start with the same letter. Hope this helps.

Question 7

In the "Appendix: All trig ratios in the unit circle" above, why is the Tangent at a negative slope?

Accepted Answer

Normally the unit circle is pictured with the hypotenuse going from the origin to the point where the ray intersect the circle (meaning that the right-angle corner in on the X axis) whereas the one pictured here runs through point of intersection to the X and Y axis outside of the unit circle. Pictured this way, it allows for one to see all the trig function versus just the "Big 3" (sin, cos, and tan).

That being said. The way this hypotenuse is oriented gives it a negative slope.

That being said.  The way this hypotenuse is oriented gives it a negative slope.

Question 8

what's an easy way to remember that sin = y-coordinate, cos = x-coordinate, and tan = y-coordinate/x-coordinate??

Accepted Answer

You could use basic trigonometry if you forget using 90 degree triangle the related angle have r as 1 and defining the point (x, y) where the ray

Question 9

Aside from rote memory, how would I be able to derive that sinθ equals to the Y coordinate or cosθ equals to the X coordinate?

I feel like the only thing I would remember instinctively would be the mnemonic "SOH CAH TOA" ... would I be able to derive it from my understanding of that?

Accepted Answer

Yes. If you know SOHCAHTOA and the unit circle, then you can derive it.

The unit circle has a radius of one, so the hypotenuse is 1. Sine is opposite over hypotenuse. Since the hypotenuse is 1, sine on the unit circle is the opposite side. When you look at the unit circle, the opposite side is perpendicular to the x-axis. This means that, essentially, the opposite side is the height from, or the distance from, the x-axis, which is the y-value.

Algebra II (2018 edition)

Unit 9: Lesson 2

Trig unit circle review

What is the unit circle definition of the trigonometric functions?

Appendix: All trig ratios in the unit circle

Check your understanding

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