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Mastering Trigonometric Identities: Cosine Difference and Exact Values

This comprehensive guide explores key concepts in trigonometry, including the derivation of cosine difference identities and methods for finding exact trigonometric function values. Learn how to apply the cosine difference identity, understand cofunction identities, and transform expressions for practical applications. The lesson includes examples with angles such as 15°, 75°, and 87°, as well as exercises to practice finding angles with respect to given trigonometric ratios. Perfect for students looking to deepen their understanding of trigonometric identities.

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Mastering Trigonometric Identities: Cosine Difference and Exact Values

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  1. Warm Up Verify the following trig identity cos2x.(tan2x + 1) = 1

  2. Trig Game Plan Date: 12/04/13

  3. How to Derive the Difference Identity for Cosine Point Q is on the unit circle, so the coordinates of Q are (cos B, sin B). The coordinates of S are (cos A, sin A). The coordinates of R are (cos(A – B), sin (A – B)).

  4. How to Derive the Difference Identity for Cosine Since the central angles SOQ and POR are equal, PR = SQ. Using the distance formula, we have

  5. How to Derive the Difference Identity for Cosine Square both sides and clear parentheses: Rearrange the terms:

  6. How to Derive the Difference Identity for Cosine Subtract 2, then divide by –2:

  7. How to Derive the Difference Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A– (–B) and use the identity for cos(A–B). Cosine difference identity Negative angle identities

  8. Cosine of a Sum or Difference ± cos (A B) = cos A cos B sin A sin B ±

  9. We do it FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 15.

  10. We Do FINDING EXACT COSINE FUNCTION VALUES Find the exact value of each expression.

  11. You Do 2gether FINDING EXACT COSINE FUNCTION VALUES Find the exact value of

  12. You Do 2gether FINDING EXACT COSINE FUNCTION VALUES

  13. We do it FINDING EXACT COSINE FUNCTION VALUES cos 87cos 93 – sin 87sin 93 You Do 2gether

  14. Cofunction Identities Similar identities can be obtained for a real number domain by replacing 90 with Confunctions are Complementary!

  15. USING COFUNCTION IDENTITIES TO FIND θ We do it Find an angle that satisfies each of the following: (a) cot θ = tan 25° tan (90°- θ) = tan 25° (b) sin θ = cos ( –30°)

  16. USING COFUNCTION IDENTITIES TO FIND θ You Do 2gether Find an angle that satisfies each of the following:

  17. USING COFUNCTION IDENTITIES TO FIND θ We Do Find an angle that satisfies each of the following:

  18. USING COFUNCTION IDENTITIES TO FIND θ You Do 2gether

  19. Time of Duty (TOD) Find the exact value of cos 75. 1. 2. Find the exact value of cos 87cos 42 + sin 87sin 42. 3. Find an angle that satisfies each of the following: tan θ = cot 64°

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