Question

1. Use the product-sum identities to write the product cos(5π)sin(8π) as a sum. Show every step and explain what you are doing in each step.

Answer 1

`\cos (5 \pi) \sin (8 \pi)=(1)/(2)[\sin 13 \pi+\sin (3 \pi)]` is the answer.

Step-by-step explanation:

To write the product
`\cos (5 \pi) \sin (8 \pi)` as the sum using product-sum identities.

The product-sum identity for
`cos A sinB` is given by

`\cos A \sin B=(1)/(2)[\sin (A+B)-\sin (A-B)]`

Now, we shall substitute the value for A and B in this formula.

Thus,
`A=5 \pi` and
`B=8 \pi`, we have,

`\cos (5 \pi) \sin (8 \pi)=(1)/(2)[\sin (5 \pi+8 \pi)+\sin (5 \pi-8 \pi)]`

Adding the terms within the bracket,

`\cos (5 \pi) \sin (8 \pi)=(1)/(2)[\sin 13 \pi-\sin (-3 \pi)]`

Since, we know that
`\sin (-x)=-\sin (x)`, we have,

`\cos (5 \pi) \sin (8 \pi)=(1)/(2)[\sin 13 \pi+\sin (3 \pi)]`

Thus, the solution is
`\cos (5 \pi) \sin (8 \pi)=(1)/(2)[\sin 13 \pi+\sin (3 \pi)]`