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Understanding Sum and Difference Identities in Trigonometry

Learn how to derive and apply Sum and Difference Identities for cosine, sine, and tangent functions in trigonometry. Discover how these identities can help find angles and segments on a Unit Circle. Practice problems from Chapter 7, Section 3, pages 442, questions 14-22.

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Understanding Sum and Difference Identities in Trigonometry

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  1. Chapter 7 Section 3 – Sum and Difference Identities

  2. Sum/Difference of Cosine • Derivation – to find the length of a segment between 2 points on a Unit Circle formed by 2 angles, you can solve using Law of Cosines • Which gives: 2 – 2cos(a – b) Or by using the distance formula, which gives: 2 – 2(cos(a)cos(b)+sin(a)sin(b) So these two must be equal! 2-2cos(a-b) = 2-2(cos(a)cos(b)+sin(a)sin(b) So… cos(a-b) = cos(A)cos(b) + sin(a)sin(b)

  3. Sum/Difference of Cosine • Given: cos(a-b) = cos(A)cos(b) + sin(a)sin(b) Then Likewise, the sum cos(a+b) (can get the same way, just replacing b with –b) cos(a+b) = cos(a)cos(b) – sin(a)sin(b)

  4. Sum/Difference of Sine • Using the phase shift difference between sine and cosine, and the previous identity, we can derive sum/difference of sine function • Sin(a±b)=sin(a)cos(b)± cos(a)sin(b)

  5. Sum/Difference of Tan • Plug sin and cos identities into ratio of Tangent. You get: • Tan(a±b) =

  6. Sum/Difference Identities • You can use these identities along with your knowledge of special angles to find other angles derived from them!

  7. Assignment • Chapter 7, Section 3 pgs442 #14 – 22 even

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