1. 7-3: Sum and Difference Identities

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

3. 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!

4. 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 ? ?

5. 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!!!

6. 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!

7. 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
= 63/65

8. 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!!!

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

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

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