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

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

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

2. Using the right triangle diagram, write an equation for y\begin{align*}y\end{align*} in terms of θ\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 x\begin{align*}x-\end{align*}value is cos(x)\begin{align*}\cos(x)\end{align*} and the y\begin{align*}y-\end{align*}value is sin(x)\begin{align*}\sin(x)\end{align*}.

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

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

5. Apply your knowledge of 306090\begin{align*}30-60-90\end{align*} right triangles and identify the coordinates of point P\begin{align*}P\end{align*}.

6. Again, using your knowledge of 306090\begin{align*}30-60-90\end{align*} right triangles, identify the coordinates of point Q\begin{align*}Q\end{align*}.

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

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

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

10. The sine of 60\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 2nd+\begin{align*}2^{nd} +\end{align*} [ANGLE]; and then press ENTER.

11. Using your knowledge of 454590\begin{align*}45-45-90\end{align*} right triangles, identify the coordinates of point R\begin{align*}R\end{align*}. _______

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

13. The sine of 45\begin{align*}45^\circ\end{align*} is ________.

## Problem 2 – Extending the Pattern

Identify the coordinates of the following points in terms of a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}.

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

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

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

Identify the measure of the following angles.

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

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

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

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