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## Algebra II (2018 edition)

### Unit 9: Lesson 2

The unit circle definition of sine, cosine, and tangent

# Trig unit circle review

Review the unit circle definition of the trigonometric functions.

## What is the unit circle definition of the trigonometric functions?

The unit circle definition allows us to extend the domain of sine and cosine to all real numbers. The process for determining the sine/cosine of any angle theta is as follows:
1. Starting from left parenthesis, 1, comma, 0, right parenthesis, move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive x-axis is equal to theta.
2. sine, left parenthesis, theta, right parenthesis is equal to the y-coordinate of your point, and cosine, left parenthesis, theta, right parenthesis is equal to the x-coordinate.
A unit circle on an x y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches (one, zero), (zero, one), (negative one, zero), and (zero, negative one). Point (cosine of theta, sine of theta) is located near one thirty o'clock on the circle. A line segment extends from (zero, zero) to the point. An angle from the arc starting at (one, zero) and opening counter clockwise to the point has an angle measurement of theta degrees. A dashed line extends vertically from the point to the positive x axis. The length from the pont to the x axis is sine of theta. The length from where the dashed line touched the x axis to (zero, zero) is cosine of theta.
The other trigonometric functions can be evaluated using their relation with sine and cosine.
Want to learn more about the unit circle definition? Check out this video.

## Appendix: All trig ratios in the unit circle

Use the movable point to see how the lengths of the ratios change according to the angle.

## Check your understanding

Problem 1
A unit circle on an x y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches (one, zero), (zero, one), (negative one, zero), and (zero, negative one). Point (sixty four hundredths, seventy seven hundredths) is located near one thirty o'clock on the circle. A line segment extends from (zero, zero) to the point. An angle from the arc starting at (one, zero) and opening counter clockwise to the point has an angle measurement of fifty degrees.
sine, left parenthesis, 50, degrees, right parenthesis, equals

## Want to join the conversation?

• In the appendix, can someone explain to me WHY cotangent, tangent, secant, and cosecant are represented in that way?
• 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)
• 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?
• 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.
• In the Appendix (moving point example above) I have difficulty relating the tangent (in blue) to a line drawn from the center of the circle through the green dot. The tangent would only match if the x-intercept of the sine (red line) was equal on both sides of the red line.
• 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?
• 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 unchangedIF (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
• Do you have a full picture of a unit circle... please help with this :)
• Why and how are trigonometric ratios related to circles?
• 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 .

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.
• So if sine, cosine, and tangent are SOH CAH TOA, could we say that secant, cosecant, and cotangent are SHO CHA TA0?
• 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.
• Why is it that you don't use the actual angles when finding tangent