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\theta is as follows:
  1. Starting from (1,0)(1,0), move along the unit circle until the angle that is formed between your position, the origin, and the positive xx-axis is equal to θ\theta.
  2. sin(θ)\sin(\theta) is equal to the yy-coordinate of your point, and cos(θ)\cos(\theta) is equal to the xx-coordinate.
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
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}