Unizor - Trigonometry - Sum of Angles Problems 5 - Misc Identities
Channel: Zor Shekhtman on UNIZOR Education
Date: 2013-10-30

The problems below use the formulas for a sine or cosine of a sum or difference of two angles.

1. Convert the following product of two trigonometric functions into a sum or difference of trigonometric functions with different arguments
cos(α)·cos(β)
Hint: Consider cos(α+β) and cos(α−β)

2. Convert the following product of two trigonometric functions into a sum or difference of trigonometric functions with different arguments
sin(α)·sin(β)

3. Convert the following product of two trigonometric functions into a sum or difference of trigonometric functions with different arguments
sin(α)·cos(β)

4. Convert the following sums of two trigonometric functions into products of trigonometric functions with different arguments
sin(α)+sin(β)
cos(α)+cos(β)
Hint: Any two values, A and B, can be represented in a form A=U+V and B=U−V where U=(A+B)/2 and V=(A−B)/2.

5. Convert the following sum into a product of trigonometric functions
S=sin(x)+sin(2x)+sin(3x)+sin(4x)
Hint: Multiply and divide the whole sum by sin(x/2) and convert each product of two sines into a difference as in the problem #2 above. After reduction you will have only two elements of a sum left, which, in turn, can be converted into a product.