Express 3 cos x + 2cos(x – 60°) in the form Rcos(x – α), where R > 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.

Mark scheme

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

1. [B1] Expand cos(x-60°) correctly and evaluate 3 cos x + 2 cos(x - 60°) to obtain 4cosx + √3sinx or unsimplified coefficients; Need to see Acosx + B sinx with A and B correct A may be 4 or 3 + 2 cos 60 and B may be √3 or 2 sin 60.
2. [B1] State R = √19 [R cos α= 4 R sin α= √3 ]; Follow through their 4 and √3. If coefficients are 3 and 2 then B0. R = √19 from R = 4.36 B0 but 4.36 seen after √19 ISW.
3. [M1] Use correct trig formulae for their expansion to find α; e.g. α = tan⁻¹(√3/4) or cos⁻¹(4/√19) or sin⁻¹(√3/√19)
4. [A1] Obtain α = 23.41°; Allow if x instead of α.
5. [B1 FT] cos⁻¹(2.5/R); SOI [55.0°]. Follow through their √19 .
6. [M1] Use a correct method to find a value of 2θ (not x) in the interval. Allow sign error in moving α to right side; 2θ = cos⁻¹(2.5/R) + 23.41° or 2θ = 360° - cos⁻¹(2.5/R) + 23.41° with R substituted.
7. [A1] Obtain one correct answer e.g. 39.2°; If working for M1 not seen then M1 implied by 39.2° or 164.2° Must be at least 1d.p.
8. [A1] Obtain second correct answer e.g. 164.2° and no others in the interval; Must be at least 1d.p. Ignore answers outside the given interval.