a sin x + b cos x = R sin (x + alpha)
a sin x - b cos x = R sin (x - alpha)
a cos x - b sin x = R cos (x + alpha)
a cos x + b sin x = R cos (x - alpha)
where R = sqrt(a2 + b2)
alpha = tan-1 (b/a)
Exercises for students:
Express 3 sin x + 4 cos x in the form R sin (x + a), where R is positive and a is acute, giving the value of a to the nearest 0.1°.
Express 5 cos x - 12 sin x in the form R cos (x + a), where R > 0 and a is acute.
Express 5 cos x + 6 sin x in the form R cos (x - a), where R > 0 and a is acute.