How do you solve #sin x - 2 cos x = 2#?

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Answer 1 · 2015-05-08T05:18:16

Call y the arc that tan y = 2 = tan 63.43 deg.
sin y = sin 63.43 = 0.89
cos y = cos 63.43 = 0.45

f(x) = #sin x - (sin y/cos y) cos x = 2#

sin x.cos y - sin y.cos x = 2.cos y

sin(x - y) = 2.cos y = 0.90

sin (x - 63.43) = sin 64.16. There are 2 answers:

a. x - 63.43 = 64.16 -> x = 128 deg

b. x - 63.43 = 180 - 64.16 = 115.84-> x = 179.27

Check:
#x = 128 -> f(x) = 0.78 - 2.(0.61) = 2# Correct
#x = 179.27 -> f(x) = 0.01 - 2.(-0.999) = 0.01 + 1.99 = 2 # Correct