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Mastering Sum and Difference Formulas in Trigonometry

This section covers the essential sum and difference formulas for sine, cosine, and tangent functions, which will be provided on the test. Key points include consistent signs for sine and tangent, while cosine shows opposite signs. Examples include calculating sin(75°) and cos(45°) to express these functions using sum or difference formulas. Additionally, we explore converting complex trigonometric expressions into algebraic forms, and how to find solutions within specified intervals. Practice questions enhance understanding for better test preparation.

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Mastering Sum and Difference Formulas in Trigonometry

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  1. Section 5.4 Sum and Difference Formulas These formulas will be given to you on the test.

  2. Sine Formulas Signs are the same

  3. Cosine Formulas Signs are opposite

  4. Tangent Formulas Signs are the same Signs are the same Opposite signs

  5. Example 1 Find the exact value for sin 75° using a sum or difference formula.

  6. sin 75° = sin (45° + 30°) = sin 45° cos30° + cos45° sin 30°

  7. Example 2

  8. Example 3

  9. Example 4 = cos (25° + 20°) = cos 45° Find the exact value of cos 25° cos 20° − sin 25° sin 20°

  10. Example 5 Write sin(arctan 1 + arccosx) as an algebraic expression.

  11. This expression fits the formula sin (u + v). Angles u = arctan 1 and v= arccosx. sin (u + v) = sin ucosv + cosu sin v = sin(arctan 1) cos(arccosx) + cos(arctan 1) sin(arccosx) 1 1 v x u 1

  12. sin (u + v) = sin(arctan 1) cos(arccosx) + cos(arctan 1) sin(arccosx) 1 1 v x u 1

  13. Example 6 In the interval [0, 2π) find all of the solutions of

  14. Example 7 Verify the identity.

  15. HW: pp 404-405 (8, 16,20-72 mult. of 4)

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