A-LevelMathematicsTrigonometry9709/12 · Feb/Mar 2022 · Q77 Marks
Show that (sin θ + 2 cos θ)/(cos θ - 2 sin θ) - (sin θ - 2 cos θ)/(cos θ + 2 sin θ) = 4/(5 cos² θ - 4)
- (a)Show that (sin θ + 2 cos θ)/(cos θ - 2 sin θ) - (sin θ - 2 cos θ)/(cos θ + 2 sin θ) = 4/(5 cos² θ - 4)[4]
- (b)Hence solve the equation (sin θ + 2 cos θ)/(cos θ - 2 sin θ) - (sin θ - 2 cos θ)/(cos θ + 2 sin θ) = 5 for 0° < θ < 180°.[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.
- [*M1] (sin θ+2cosθ)(cos θ + 2 sin θ) – (sin θ – 2 cos θ)(cos θ – 2sinθ) / (cosθ-2sinθ) (cosθ+2sinθ)
- [A1] 5 sin θ cos θ + 2cos² θ+2sin² θ-(5sin θ cos θ-2sin² θ-2cos² θ) / cos² θ-4 sin² θ
- [DM1] 4(cos²+sin²θ) / cos² θ-4 sin² θ
- [A1 AG] 4 / cos² θ-4(1-cos²)
- [M1] 4 / (5 cos² θ – 4) = 5 leading to 25 cos² θ = 24
- [A1] leading to cos θ = √(24/25) [= (±)0.9798]
- [A1 FT] θ = 11.5° or 168.5°
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