Part b) most likely has a typo, where the second sin θ should be a cos θ instead

If interpreted as printed, the sin θ could be factorised to give :


(√5 - 2) sin θ = 1

sin θ = 1/(√5 - 2)

sin θ = (√5 + 2)/[(√5 - 2)(√5 + 2)]

(Rationalising the denominator)

sin θ = (√5 + 2)/((√5)² - 2²)

(From the property (a + b)(a - b) = a² - b² )

sin θ = √5 + 2 (not possible as |sinθ| ≤ 1)

The correct one should be :

√5 sin θ - 2 cos θ = 1

Use the R-formula variant :

a sin θ ± bcos θ = R sin (θ ± α), whereby

R = √(a² + b²)
tan α = b/a
a,b > 0 and α is acute.



R = √((√5)² + 2²) = √(5 + 4) = √9 = 3

tan α = 2/√5
α = tan-¹ (2/√5) = 41.81° (2d.p)

So,


√5 sin θ - 2 cos θ = 3 sin (θ - 41.81°) = 1

sin (θ - 41.81°) = ⅓ (positive, look at 1st and 2nd quadrants)

θ - 41.81° = sin-¹ (⅓)

Basic angle = 19.47° (2d.p)

θ - 41.81° = -180° - 19.47°, 19.47°

θ = -180° - 19.47° + 41.81°, 19.47° + 41.81°

θ = -157.66°, 61.28°

θ = -157.7°, 61.3° (1d.p)