# 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, \begin{align*}\theta\end{align*}, and the coordinates of point \begin{align*}P\end{align*}, a corresponding point on the circle.

*Note that the angle \begin{align*}\theta\end{align*} is measured from the positive \begin{align*}x-\end{align*}axis.*

Right triangle trigonometry and knowledge of special right triangles can be applied to understanding the relationship between \begin{align*}\theta\end{align*} and \begin{align*}P\end{align*}. (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 \begin{align*}x\end{align*} in terms of \begin{align*}\theta\end{align*}.

2. Using the right triangle diagram, write an equation for \begin{align*}y\end{align*} in terms of \begin{align*}\theta\end{align*}.

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

3. What is the value of \begin{align*}a\end{align*} when the hypotenuse is 1 unit?

4. What is the value of \begin{align*}b\end{align*} when the hypotenuse is 1 unit? Don’t forget to rationalize the denominator!

5. Apply your knowledge of \begin{align*}30-60-90\end{align*} right triangles and identify the coordinates of point \begin{align*}P\end{align*}.

6. Again, using your knowledge of \begin{align*}30-60-90\end{align*} right triangles, identify the coordinates of point \begin{align*}Q\end{align*}.

7. The cosine of \begin{align*}30^\circ\end{align*} is ________.

8. The sine of \begin{align*}30^\circ\end{align*} is ________.

9. The cosine of \begin{align*}60^\circ\end{align*} is ________.

10. The sine of \begin{align*}60^\circ\end{align*} is ________.

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

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

11. Using your knowledge of \begin{align*}45-45-90\end{align*} right triangles, identify the coordinates of point \begin{align*}R\end{align*}. _______

12. The cosine of \begin{align*}45^\circ\end{align*} is ________.

13. The sine of \begin{align*}45^\circ\end{align*} 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 \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.

14. \begin{align*}T\end{align*} __________

15. \begin{align*}U\end{align*} __________

16. \begin{align*}V\end{align*} __________

Identify the measure of the following angles.

17. \begin{align*}m \angle WOT = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}m \angle WOU = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}m \angle WOV = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

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