Question

Find all values of `x` in `[0^circ, 360^circ)` to satisfy this equation? `sin x + sqrt3= 3 sqrt3 cos x`

Answer 1

`59^@83; 278^@37`

Explanation:

`sin x + sqrt3 = 3sqrt3cos x`
`sin x - 3sqrt3cos x = - sqrt3` (1)
Call `tan t = sin t/cos t = 3sqrt3 = 5.196` -->
`t = 79^@10`, and cos t = 0.19
Equation (1) becomes:
`sin x.cos t - sin t.cos x = - sqrt3cost = - 0.33`
`sin (x - 79^@10) = - 0.33`
Calculator and unit circle give 2 solutions for (x - 79.10):

a. `x - 79.10 = - 19^@27`
`x = - 19.27 + 79.10 = 59^@83`
b. `x - 79.10 = 180 - (-19.27) = 199^@27`
`x = 199.27 + 79.10 = 278^@37`
Check by calculator.
x = 59.83 --> sin x = 0.86 --> `sin x + sqrt3 = 2.60` -->
cos x = 0.50 -->`3sqrt3.cos x = 2.60`. Proved.
x = 278.37 --> sin x = - 0.99 --> `sin x + sqrt3 = 0.74`
cos x = 0.14 --> `3sqrt3cos x = 0.75`. Proved