Question #81bd2

1 Answer
Nghi N.
Aug 23, 2015

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

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