Express `3sinx + 2√2cos(x + (1/4)π)` in the form `Rsin(x + α)`, where `R > 0` and `0 < α < (1/2)π`. State the exact value of R and give α correct to 3 decimal places.

Mark scheme

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

1. [B1] Use the correct expansion of cos(x + (1/4)π) to obtain sinx + 2cosx
2. [B1 FT] State R = √5
3. [M1] Use correct trig formulae to find α
4. [A1] Obtain α =1.107
5. [B1 FT] sin( (1/2)θ + α ) = 1.5 / R
6. [M1] Use a correct method to obtain an un-simplified value of θ with their α
7. [A1] Obtain one correct answer e.g. -0.74 in the interval
8. [A1] Obtain second correct answer e.g. 2.60 (2.5986) or 4π – 0.74 = 11.8 or 2.60 – 4π = -9.97 in the interval
9. [A1] Obtain two more correct answers e.g. -9.97 and 11.8 and no others in the interval