Question

Solve 4sinx-cosx+3=0 ?

Answer 1

`x = 240^@93 + k360^@`
`x = 327^@12 + k360^@`

Explanation:

4sin x - cos x = -3
Divide both sides by 4 -->
`sin x - (1/4)cos x = - 3/4` (1)
Call `tan t = sin/(cos t) = 1/4` --> `t = 14^@04` -->
`cos t = 0.97`.
The equation (1) becomes:
`sin x.cos t - sin t.cos x = -(3/4)cos t = -(3/4)(0.97) = - 0.73`
`sin (x - t) = sin (x - 14.04) = - 0.73`
Calculator and unit circle give 2 solutions for (x - 14.04) -->
a. `x - 14.04 = - 46.89`
x = - 46.89 + 14.04 = - 32.85, or
`x = 327^@15` (co-terminal).
b. `x - 14.04 = 180 - (-46.89) = 226^@89`
`x = 226.89 + 14.04 = 240^@93`
For general answers, add `k360^@`
Check by calculator.
x = 240.93 --> 4sin x = - 3.50 --> cos x = - 0.50
4sin x - cos x + 3 = - 3.50 + 0.50 + 3 = 0. Proved.