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Course: Algebra 2 > Unit 11

Lesson 1: Unit circle introduction
Math
Algebra 2
Trigonometry
Unit circle introduction
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Trig unit circle review

Google Classroom
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 θ is as follows:
  1. Starting from (1,0), 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 θ.
  2. sin⁡(θ) is equal to the y-coordinate of your point, and cos⁡(θ) 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.
sin⁡(50∘)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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