AQA A-Level Maths: Trigonometry

The function $\text{f}$f is defined by $\text{f}(\theta) = 7\sin\theta + 24\cos\theta,$f(θ)=7sinθ+24cosθ, where $\theta$θ is measured in degrees.

(a) Express $\text{f}(\theta)$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 $7\sin\theta + 24\cos\theta = 12.5$7sinθ+24cosθ=12.5 for $0 \leq \theta < 360^{\circ}$0≤θ<360∘, giving your answers to one decimal place. (5 marks)

(c) The function $\text{g}$g is defined by $\text{g}(\theta) = \frac{50}{7\sin\theta + 24\cos\theta + 30}.$g(θ)=7sinθ+24cosθ+3050​. Find the maximum value of $\text{g}(\theta)$g(θ) and the value of $\theta$θ in the interval $0 \leq \theta < 360^{\circ}$0≤θ<360∘ at which it 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. (3 marks)

(b) Hence solve the equation $\cos 2\theta + \sin\theta = 0$cos2θ+sinθ=0 for $0 \leq \theta < 2\pi$0≤θ<2π, giving your answers in terms of $\pi$π. (7 marks)

A sector $OAB$OAB of a circle has centre $O$O, radius $10\ \text{cm}$10 cm and angle $AOB = 1.2$AOB=1.2 radians. The chord $AB$AB divides the sector into the 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 your answer to three significant figures. (5 marks)

(Work in radians throughout.)

Solve the equation $\csc^2 x + \cot x = 7$csc2x+cotx=7 for $0 \leq x < 360^{\circ}$0≤x<360∘, giving your answers to one decimal place. (6 marks)

Find the exact value of $\sin\!\left(\arctan\tfrac{3}{4} + \arccos\tfrac{5}{13}\right),$sin(arctan43​+arccos135​), giving your answer as a fraction in its simplest form. In your solution, comment briefly on why both inverse-trig values may be taken as acute angles. (5 marks)

When $x$x is small and measured in radians, use the small-angle approximations for $\sin$sin, $\cos$cos and $\tan$tan to show that $\frac{1 - \cos 4x}{x\tan 2x} \approx 4.$xtan2x1−cos4x​≈4. (4 marks)