Express 3 sin 2θ sec θ + 10 cos(θ – 30°) in the form R sin(θ + α) where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] Use sin 2θ = 2 sin θ cos θ and secθ = 1/cos θ to obtain 6sin θ
2. [B1] Expand second term to obtain 5√3 cos θ + 5 sin θ
3. [B1] Simplify to obtain 11 sin θ + 5√3 cos θ
4. [B1 FT] State R=14. FT their k₁ cosθ + k₂ sinθ
5. [M1] Use appropriate trigonometry to find α
6. [A1] Obtain α = 38.21 AWRT
7. [B1 FT] State or imply 14sin(2β+38.21) = 2. FT their R and α
8. [M1] Carry out correct process to find value of β between 0° and 90°
9. [A1] Obtain 66.8 AWRT