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, and the coordinates of point P, a corresponding point on the circle.
Note that the angle θ is measured from the positive x−axis.
Right triangle trigonometry and knowledge of special right triangles can be applied to understanding the relationship between θ 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 θ.
2. Using the right triangle diagram, write an equation for y in terms of θ.
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∘ is ________.
8. The sine of 30∘ is ________.
9. The cosine of 60∘ is ________.
10. The sine of 60∘ 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 2nd+ [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∘ is ________.
13. The sine of 45∘ 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.