A-LevelMathematicsTrigonometry9709/21 · May/June 2023 · Q711 Marks

Express 7 cos θ + 24 sin θ in the form Rcos(θ – α), where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.

(a)Express 7 cos θ + 24 sin θ in the form Rcos(θ – α), where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.[3]
(b)Solve the equation 7 cos θ + 24 sin θ = 18 for 0° < θ < 360°.[4]
(c)As β varies, the greatest possible value of 150 / (7 cos ½β + 24 sin ½β + 50) is denoted by V. Find the value of V and determine the smallest positive value of β (in degrees) for which the value of V occurs.[4]

Mark scheme

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

[B1] State R = 25
[M1] Use appropriate trigonometry to find α Allow if found in radians.
[A1] Obtain α = 73.74 or greater accuracy.
[M1] Use correct method to find one value of θ
[A1] Obtain 29.8 (or 117.7) or greater accuracy.
[M1] Use correct method to find second value of θ between 0 and 360
[A1] Obtain 117.7 (or 29.8) or greater accuracy; and no others between 0 and 360.
[B1 FT] State or imply expression is 150 / (25 cos(½β - 73.74) + 50) following their R and α.
[B1] Obtain V = 6
[M1] Attempt complete method to find positive value from cos(½β−73.74) = -1 for their α.
[A1] Obtain 507.5 or greater accuracy.

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