Edexcel A-Level Maths: Trigonometry

The function $f$f is defined by $f(\theta) = 5\cos\theta + 12\sin\theta$f(θ)=5cosθ+12sinθ, where $\theta$θ is measured in degrees.

(a) Express $f(\theta)$f(θ) in the form $R\cos(\theta - \alpha)$Rcos(θ−α), where $R > 0$R>0 and $0 < \alpha < 90^{\circ}$0<α<90∘. Give the exact value of $R$R and the value of $\alpha$α to one decimal place. (4 marks)

(b) Hence solve the equation $5\cos\theta + 12\sin\theta = 6.5$5cosθ+12sinθ=6.5 for $0 \leq \theta < 360^{\circ}$0≤θ<360∘, giving your answers to one decimal place. (5 marks)

(c) State the maximum value of $f(\theta)$f(θ) and the smallest positive value of $\theta$θ at which this maximum occurs. (3 marks)

(a) Starting from the addition formula for $\cos(A + B)$cos(A+B), prove the double-angle identity

$\cos 2\theta \equiv 1 - 2\sin^2\theta$cos2θ≡1−2sin2θ

(4 marks)

(b) Hence solve the equation

$\cos 2\theta + 3\sin\theta - 2 = 0$cos2θ+3sinθ−2=0

for $0 \leq \theta < 2\pi$0≤θ<2π, giving your answers in terms of $\pi$π. (6 marks)

The diagram referred to below shows a sector $OAB$OAB of a circle of centre $O$O and radius $8\ \text{cm}$8 cm. The angle $AOB$AOB is $0.75$0.75 radians. The chord $AB$AB divides the sector into a triangle $OAB$OAB and a shaded segment.

(a) Find the length of the arc $AB$AB and hence the perimeter of the sector $OAB$OAB. (3 marks)

(b) Find the area of the sector $OAB$OAB and hence the area of the shaded segment, giving each answer to 3 significant figures. (5 marks)

(Work in radians throughout.)

Solve the equation

$2\tan^2 x + \sec x = 1$2tan2x+secx=1

for $0 \leq x < 360^{\circ}$0≤x<360∘, giving your answers to one decimal place where appropriate. (6 marks)

Prove that

$\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta} \equiv 2\sec^2\theta$1−sinθ1​+1+sinθ1​≡2sec2θ

(5 marks)

When $\theta$θ is small and measured in radians, use the small-angle approximations for $\sin$sin and $\cos$cos to show that

$\frac{\cos 3\theta - 1}{\theta\sin 2\theta} \approx -\frac{9}{4}$θsin2θcos3θ−1​≈−49​

(4 marks)