# 2.2: Round and Round She Goes

**At Grade**Created by: CK-12

*This activity is intended to supplement Trigonometry, Chapter 1, Lesson 7.*

## Problem 1 – Introduction to the Unit Circle

To the right, you will see a special circle known as the unit circle. It is centered at the origin and has a radius of one unit.

This circle is very important to the field of trigonometry. It is essential to develop an understanding of relationships between the angle theta, , and the coordinates of point , a corresponding point on the circle.

*Note that the angle is measured from the positive axis.*

Right triangle trigonometry and knowledge of special right triangles can be applied to understanding the relationship between and . (Note that the hypotenuse of this triangle is 1 unit, corresponding to the radius of 1 unit on the unit circle.)

1. Using the right triangle diagram, write an equation for in terms of .

2. Using the right triangle diagram, write an equation for in terms of .

Using the answers to Exercises 1 and 2, the unit circle can be relabeled as shown to the right. Note that the value is and the value is .

3. What is the value of when the hypotenuse is 1 unit?

4. What is the value of when the hypotenuse is 1 unit? Don’t forget to rationalize the denominator!

5. Apply your knowledge of right triangles and identify the coordinates of point .

6. Again, using your knowledge of right triangles, identify the coordinates of point .

7. The cosine of is ________.

8. The sine of is ________.

9. The cosine of is ________.

10. The sine of is ________.

Check your results to Exercises 7–8 using your graphing calculator as shown to the right.

Note the symbol can be found by pressing **[ANGLE]**; and then press **ENTER**.

11. Using your knowledge of right triangles, identify the coordinates of point . _______

12. The cosine of is ________.

13. The sine of is ________.

Check your results to Exercises 11–13 using your graphing calculator.

## Problem 2 – Extending the Pattern

Identify the coordinates of the following points in terms of and .

14. __________

15. __________

16. __________

Identify the measure of the following angles.

17.

18.

19.