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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 ________.

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 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{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}

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Feb 23, 2012

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Nov 04, 2014
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TI.MAT.ENG.SE.1.Trigonometry.2.2

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