Express 7 sinθ+24 cosθ in the form Rcos(θ-α), where R > 0 and 0 < α < ½π. Give the value of α correct to 4 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] State R = 25 From correct work. BOD if correct value follows use of their α, but not if a decimal approximation to R is seen first.
2. [M1] Use correct trig formula to find α, e.g. α = tan^-1(7/24) If cos α = 24 and sin α = 7 seen, then M0 A0.
3. [A1] Obtain α = 0.2838 CAO
4. [B1FT] cos^-1(24.5/25) Can be implied by ±0.2(0033...), or by x = 0.08... or 0.48... or a correct value of x following M1. FT their R from (a). Allow B1 if cos^-1(24.5/R) is not evaluated, provided R≥24.5.
5. [M1] Carry out a complete correct method to find a value of x Sight of a correct equation using (a) is needed, e.g. cos(x-0.2838) = 0.98 OE
6. [A1] Obtain answer (x = ) 1.45 AWRT Need 3 sf or better.
7. [A1] Obtain (x = ) 0.250 or (x = ) 0.251, and no other in the interval AWRT Ignore answers outside the given interval.