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# 2.2: Round and Round She Goes

Difficulty Level: 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, $\theta$, and the coordinates of point $P$, a corresponding point on the circle.

Note that the angle $\theta$ is measured from the positive $x-$axis.

Right triangle trigonometry and knowledge of special right triangles can be applied to understanding the relationship between $\theta$ and $P$. (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 $x$ in terms of $\theta$.

2. Using the right triangle diagram, write an equation for $y$ in terms of $\theta$.

Using the answers to Exercises 1 and 2, the unit circle can be relabeled as shown to the right. Note that the $x-$value is $\cos(x)$ and the $y-$value is $\sin(x)$.

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

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

5. Apply your knowledge of $30-60-90$ right triangles and identify the coordinates of point $P$.

6. Again, using your knowledge of $30-60-90$ right triangles, identify the coordinates of point $Q$.

7. The cosine of $30^\circ$ is ________.

8. The sine of $30^\circ$ is ________.

9. The cosine of $60^\circ$ is ________.

10. The sine of $60^\circ$ is ________.

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

Note the $^\circ$ symbol can be found by pressing $2^{nd} +$ [ANGLE]; and then press ENTER.

11. Using your knowledge of $45-45-90$ right triangles, identify the coordinates of point $R$. _______

12. The cosine of $45^\circ$ is ________.

13. The sine of $45^\circ$ is ________.

## Problem 2 – Extending the Pattern

Identify the coordinates of the following points in terms of $a$ and $b$.

14. $T$ __________

15. $U$ __________

16. $V$ __________

Identify the measure of the following angles.

17. $m \angle WOT = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

18. $m \angle WOU = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

19. $m \angle WOV = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

Feb 23, 2012

Nov 04, 2014