Question

Express f(x) = 6cos x + 2 sin x as Rcos (x + theta), where theta is an acute angle in radians.

Answer 1

We want to find R and θ such that

R cos(x + θ) = R (cos(x) cos(θ) - sin(x) sin(θ)) = 6 cos(x) + 2 sin(x)

which means

R cos(θ) = 6

-R sin(θ) = 2

Observe that

(R cos(θ))² + (-R sin(θ))² = 6² + 2² ==> R ² = 40 ==> R = 2√10

and

-R sin(θ) / (R cos(θ)) = 2/6 ==> tan(θ) = -1/3 ==> θ = arctan(-1/3) = -arctan(1/3)

So we have

6 cos(x) + 2 sin(x) = R cos(x + θ) = 2√10 cos(x - arctan(1/3))