How do you solve #Sin x - cos x = 1/3#?

1 Answer
Nghi N.
Jun 30, 2016

#58^@63 and 211^@37#

Explanation:

Call t the arc that #tan t = sin t/(cos t) = 1#. --> #t = 45^@#.
The equation can be written as:
#sin x - (sin t/(cos t))cos x = 1/3#
#sin x.cos t - sin t.cos x = (cos t)/3 = (cos 45)/3 = sqrt2/6#
#sin (x - t) = sin (x - 45) = sqrt2/6 = 0.235.#
There are 2 solutions. On the unit circle, 0.235 is the same value of both sin (13^@63) and sin (180 - 13.63) = sin 166^@37
a. #(x - 45) = 13^@63#.
#x = 13.63 + 45 = 58^@63#
b. #(x - 45) = 180 - 13.63 = 166^@37#
#x = 166.37 + 45 = 211^@37#
Answers for (0, 360):
#58^@63 and 211^@37#

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