Express 5 sin(x+π/6)-4 cosx in the form Rsin(x-α), where R > 0 and 0 < α < π/2. State the exact value of R and give the value of α correct to 3 decimal places.

Mark scheme

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

1. [B1] Expand 5 sin(x+π/6) - 4 cos x to obtain (5/2)√3 sin x - (1/2) cos x Or exact 2 term equivalent.
2. [B1FT] State R = √21 Follow their (5/2)√3 and (1/2).
3. [M1] Use correct trig formulae to obtain tan α OE, WWW. E.g. tan α = (1/2) / (5√3/2) = 1/(5√3). sin α = (1/2) / √21 = 1/(2√21). cos α = (5√3/2) / √21 = 5√3/(2√21).
4. [A1] Obtain α = 0.333
5. [B1FT] sin(2θ - α) = √7/√21 SOI 0.615... can be implied by one correct answer. Follow their √21.
6. [M1] Use a correct method to obtain a value of θ in the interval
7. [A1] Obtain one correct answer, e.g. 0.47
8. [A1] Obtain second correct answer, e.g. 1.43, and no others in the interval