Question

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

Answer 1

`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`