Question

Express sin x - 3 cos x in the form of r sin (x-y) with r>0 and 0< y < (pi/2) ?

Answer 1

`f(x) = 3.16.sin (x - 71^@57)`

Explanation:

f(x) = sin x - 3cos x
Call `tan y = sin y/(cos y) = 3` --> `y = 71^@57` and
cos y = 0.316
`f(x) = sin x - (sin y.cos x)/(cos y)`
`f(x) = (sin x.cos y - sin y.cos x)/(cos y)`
Reminder of trig identity -->
(sin x.cos y - sin y.cos x) = sin (x - y)
Therefor,
`f(x) = rsin (x - y)`, with
`r = 1/(cos y) = 1/0.316 = 3.16`, and `y = 71^@57`
Finally,
`f(x) = 3.16.sin (x - 71^@57)`