Question

Question #81bd2

Answer 1

Solve `cos (x + 30) - sin x = 1/2`

Ans: 13.22 and -133.22 deg

Explanation:

Apply the sum cosine trig identity:
`cos (x + 30) = sqrt3/2cos x - 1/2 sin x`. The main equation becomes;
`(sqrt3/2)cos x - (3/2)sin x = 1/2`
`sqrt3cos x - 3sin x = 1`. Divide both side by `sqrt3`
`cos x - sqrt3sin x = 1/sqrt3`
Replace in the left side: `sqrt3` by `tan 60 = sin 60/cos 60`, we get:
`cos x.cos 60 - sin 60.sin x = cos 60/sqrt3`
`cos (x + 60) = 1/(2sqrt3) = 0.29`. Calculator gives -->
`(x + 60) = +- 73.22` deg

a. `x + 60 = 73.22` --> `x = 73.22 - 60 = 13.32` deg
b. `x + 60 = - 73.22 --> x = -73.22 - 60 = -133.22` deg

Check by calculator:
x = 13.22 --> cos (x + 30) = 0.73 --> sin 13.22 = 0.13.
we get: 0.73 - 0.13 = 0.50. OK
x = -133.32 --> sin x = 0.73 --> cos (x + 30) = cos (-103.22) = -0.23.
We get: -0.23 + 0.73 = 0.50. OK