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 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.
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
sine, left parenthesis, 50, degree, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i