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Exact Values of Sine and Cosine for Special Angles

In this section, we explore finding exact values of sine and cosine for various angles using formulas, particularly focusing on special right triangle angles and quadrantal angles. We will review the simplified radical forms without calculators. Key angles of 30°, 45°, and 60° will be covered, along with example problems and identities. These exercises help reinforce understanding of sine and cosine values, their relationships, and their applications in solving trigonometric equations. Practice angles include both sum and difference concepts.

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Exact Values of Sine and Cosine for Special Angles

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  1. Section 10.1

  2. We will be finding exact values of sine and cosine of angles. (using formulas) • Recall trig values of special right triangle angles and quadrantal angles. • Simplified radical form (no calculators!)

  3. Cosine formulas:

  4. Sine formulas:

  5. Review: special angles 30° 2 60° 1 S A 45° 1 T C 45° 1

  6. Review: quadrantal angles (cos, sin) • (0,1) (1,0) (-1,0) • • • (0,-1)

  7. Ex. 1: Find the exact value of each expression: (express angle as sum or difference) a.) sin 75 b.) cos 15

  8. Ex. 2: Find the exact value of the expression: a.) sin 285°

  9. Ex. 3: Find the exact value of each expression: (work backwards) a.) sin 80 cos 20 - cos 80 sin 20 b.)

  10. Ex. 4:Prove that the given equation is an identity: (work on one side only)

  11. Example 5: (sketch right triangles in correct quadrants, use Pyth. Theorem where necessary) 13 5 12 3 -5 4 Do not cancel!

  12. Example 6: -12 3 (Be careful of signs!) -5 -4 13 5

  13. Example 7: Simplify: sin (60 + x) – sin (60 – x)

  14. Example 8: Simplify:

  15. Homework 3: Page 373 #2-30 even (skip #20)

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