7-3: Sum and Difference Identities

Objectives Use the sum and difference identities for the sine, cosine and tangent functions.

Sum and Difference Identities for Cosine If α and β represent the measures of two angles, then the following identities hold for all values of α and β Notice the difference!

Example Show by producing a counterexample that cos(x-y)≠cosx-cosy. Let x=π/4 and y=π/4. cos(π/4 – π/4) = cos π/4 – cos π/4 cos 0 = √2/2 – √2/2 1 ≠ 0 ? ?

Example Use the sum or difference identity to find the exact value of cos 75°. cos(75°) = cos(30°+ 45°) = cos30°cos45° – sin30°sin45° You could have also used 135° - 60°. If you use the calculator, you will get a decimal approximation!!!

Sum and Difference Identities for Sine If α and β represent the measures of two angles, then the following identities hold for all values of α and β Now the signs match!

Example Find the value of sin(x+y) if 0<x<π/2, 0<y<π/2, sinx=4/5 and siny=5/13. sin(x+y) = sinxcosy+cosxsiny = (4/5)(12/13) + (5/13)(3/5) = 48/65 + 15/65 = 63/65

Sum and Difference Identities for Tangent If α and β represent the measures of two angles, then the following identities hold for all values of α and β Notice the signs now!!!

Example Use the sum or difference identity to find the exact value of tan 255°. tan(255°) = tan(225 + 30°) You could have also used 210° + 45°. If you use the calculator, you will get a decimal approximation!!!

Example Verify that sec(π+A) = - secA is an identity.

Homework 7-3: p. 442 #15-24 multiples of 3 #26-30 #34-38