Express 5 sin θ + 12 cos θ in the form R cos(θ – α), where R > 0 and 0 < α < ½π.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

1. [B1] State R = 13
2. [M1] Use correct trig formulae to find α = tan⁻¹(±5/12) = cos⁻¹(±12/13) = sin⁻¹(±5/13)
3. [A1] Obtain α = 0.395 CWO If negative sign seen when finding R then A0 here. If degrees 22.6 A0 MR. Only penalise degrees once in (a) and (b). Note α = 0.39479...
4. [B1FT] SOI 1.0910... FT their incorrect R.
5. [M1] Use correct method to find a value of 2x in the interval
6. [A1] Obtain answer, e.g. x = 0.743 or 0.742
7. [A1] Obtain second answer, e.g. x = 2.79 and no others in the interval