A-LevelMathematicsTrigonometry9709/22 · Feb/Mar 2025 · Q710 Marks

Express 6 sin θ-4 cosθ in the form Rsin(θ-α), where R > 0 and 0° < α < 90°. Give the exact value of R and the value of α correct to 2 decimal places.

(a)Express 6 sin θ-4 cosθ in the form Rsin(θ-α), where R > 0 and 0° < α < 90°. Give the exact value of R and the value of α correct to 2 decimal places.[3]
(b)Hence solve the equation 6 sin θ-4 cosθ+5 = 0 for 0° < θ < 360°.[4]
(c)As the value of β varies, find the greatest possible value of (3 sin 4β-2 cos 4β)² + 15 and determine the smallest positive value of β, in degrees, for which this greatest value occurs.[3]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

[B1] State R = √52 or 2√13
[M1] Use appropriate trigonometry to find α
[A1] Obtain α = 33.69 Or greater accuracy.
[M1] Use correct method to find one value of θ
[A1] Obtain 257.6 (or 349.8) Or greater accuracy 257.587... (or 349.792...).
[M1] Use correct method to find second value of θ between 0 and 360
[A1] Obtain 349.8 (or 257.6) and no others between 0 and 360 Or greater accuracy.
[B1] Obtain greatest value 28
[M1] Attempt complete method to find positive value of β using sin(4β-α)=1
[A1] Obtain ¼(90+33.69) and hence 30.9 Or greater accuracy (30.922...).

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