Express 4 cosθsin(θ+30°) in the form Rcos (2θ-α)+k, where R > 0, 0° < α < 90° and k is a constant.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] Expand to obtain the form k₁ cosθsinθ + k₂ cos² θ
2. [A1] Obtain 2√3 cos θsinθ + 2cos² θ with exact coefficients
3. [A1] Obtain √3sin 2θ+cos2θ+1 FT Following their expression in sinθ and cosθ.
4. [B1] State R = 2 FT Following their expression in sin 2θ and cos2θ.
5. [M1] Use appropriate trigonometry to find value of α Must be of the form a sin 2θ+ b cos 2θ.
6. [A1] Conclude 2cos(2θ-60)+1
7. [*M1] Attempt to express equation in form cos(4φ-α)=k
8. [DM1] Use correct process to find one value of φ
9. [A1] Obtain cos(4φ-60) = 1/2 and hence 32.6 Or greater accuracy 32.6321...
10. [DM1] Use correct process to find second value of φ
11. [A1] Obtain 87.4 and no others between 0 and 90 Or greater accuracy 87.3678...