Express 4 sin θ sin(θ+ 60°) in the form a+Rsin(2θ-α), where a and R are positive integers and 0° < α < 90°.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [B1] (a) Expand to obtain 2sin² θ + 2√3sinθcos θ
2. [*M1] (a) Attempt to express in terms of sin 2θ and cos 2θ
3. [A1] (a) Obtain 1 − cos 2θ + √3sin 2θ
4. [A1 FT] (a) Obtain R = 2
5. [DM1] (a) Use appropriate trigonometry to find α
6. [A1] (a) Obtain 1+2sin(2θ-30)
7. [B1] (a) Alternative Method: Expand to obtain 2sin² θ + 2√3sin θ cos θ
8. [*M1] (a) Alternative Method: For expansion of a + Rsin(2θ – α) and comparison with 2sin² θ + 2√3sin θ cos θ
9. [DM1] (a) Alternative Method: Rsina=1 and Rcosa = √3 leading to values of R and α
10. [A1 FT] (a) Alternative Method: Obtain R = 2
11. [A1] (a) Alternative Method: Obtain α = 30° and α ≠ 1
12. [A1] (a) Alternative Method: Obtain 1+2sin(2θ-30)
13. [*B1 FT] (b) State sin(2θ-30) = -1/5
14. [DM1] (b) Attempt complete method to find smallest positive value
15. [A1] (b) Obtain 123.4 only