Section 3 – Sum and Difference Identities Chapter 7 Section 3 – Sum and Difference Identities

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)

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)

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)

Sum/Difference of Tan Tan(a±b) = (tan(a)±tan(b)) (1∓tan(a)tan(b)) Plug sin and cos identities into ratio of Tangent. You get: Tan(a±b) = (tan(a)±tan(b)) (1∓tan(a)tan(b))

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

Assignment Chapter 7, Section 3 pgs 442 #14 – 22 even