Express f (θ) = sin θ + √3 cos θ in the form r sin (θ + a) where r > 0 and 0° ≤ a ≤ 90°. At what value of theta does the minimum occur, what is the minimum value

Question

Solving trigonometric identities

Answer 1

rsin(θ+a)=rsin(θ)cos(a)+rcos(θ)sin(a)=sin(θ)+cos(θ)√3.

Therefore rcos(a)=1, rsin(a)=√3, and tan(a)=√3/1=√3, a=π/3 or 60°.

rsin(π/3)=√3, r√3/2=√3, r=2. rsin(θ+a)=2sin(θ+π/3).

The minimum value of θ is therefore π/3 or 60°.