Question

Can I apply the sum-to-product formula to `cos3theta+(sqrt2-1)costheta` and how do I do it?

I'm just wondering how the coefficient before `costheta` will affect the application of the formula.

Answer 1

Solve trig equation

Explanation:

I don't understand the question. Do you want to solve or to simplify
the expression? Yes, you can't use sum-to -product trig identity because of the coefficient.
Factor the expression:
Use trig identity:
`cos 3a = cos a(4cos^2 a - 3)`
We get:
`cos 3t + (sqt2 - 1)cos t = cos t(4cos^2 t - 3 + sqrt2 - 1) =`
`= cos t(4cos^2 t - 4+ sqrt2)`
Solve the equation, by using calculator and unit circle -->
a. cos t = 0 --> `t = pi/2`, and `t = (3pi)/2`
b. `(4cos^2 t - 4 + sqrt2) = 0`
`4cos^2 t = 4 - sqrt2 = 2.59`
`cos^2 t = 0.65`
`cos t = +- 0.80`
Calculator, and unit circle give -->
cos t = 0.80 -->`t = +- 36^@87`
`cos t = - 0.80` ---> `t = +- 143^@13`
For general answers , add `2pi = 360^@`