A-LevelMathematicsTrigonometry9709/22 · May/June 2019 · Q711 Marks

Express 4 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°.

(a)(i)Express 4 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°.[3]
(a)(ii)Hence find the smallest positive value of θ satisfying the equation 4 sin θ + 4 cos θ = 5.[2]
(b)Solve the equation 4 cot 2x = 5 + tan x for 0 < x < π, showing all necessary working and giving the answers correct to 2 decimal places.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B1] State R = √32 or equivalent or 5.657...
[M1] Use appropriate trigonometry to find α
[A1] Obtain α = 45
[M1] Carry out correct process to find one value of θ
[A1] Obtain 17.1
[B1] Use or imply cot2x = 1/tan2x
[M1] Use identity of form tan2x = (±2tanx) / (1±tan²x) to obtain equation in tanx
[A1] Obtain 6 tan²x + 10 tan x - 4 = 0 or equivalent
[M1] Attempt solution of 3-term quadratic equation for tan x
[A1] Obtain tan x = 2/3 and hence 0.32
[A1] Obtain tan x = -2 and hence 2.03 and no others between 0 and π

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.