Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The value of $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ$ is
2. If $A+B = 45^\circ$, then $(1+\tan A)(1+\tan B)$ is equal to
3. The value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7}$ is
4. The minimum value of $2 \sin^2 x + 3 \cos^2 x$ is
5. If $\tan \alpha = \frac{m}{m+1}$ and $\tan \beta = \frac{1}{2m+1}$, then $\alpha + \beta$ is
6. Find the value of $\tan(\pi/8)$.
7. Prove that $\cos 4x = 1 - 8\sin^2 x \cos^2 x$.
8. Find the value of $\sqrt{3} \csc 20^\circ - \sec 20^\circ$.
9. Prove that $\cos^2 x + \cos^2(x+\frac{\pi}{3}) + \cos^2(x-\frac{\pi}{3}) = \frac{3}{2}$.
10. If $\tan x = 3/4$ and $\pi < x < 3\pi/2$, find the values of $\sin(x/2), \cos(x/2)$ and $\tan(x/2)$.
11. Prove that $\cos 6x = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1$.
12. Case Study: The displacement of a particle executing simple harmonic motion is given by $y = 5 \sin(10t) + 12 \cos(10t)$.
    (i) Express the function in the form $R \sin(10t + \alpha)$.
    (ii) Find the maximum displacement (amplitude).
    (iii) Find the value of $y$ at $t = \pi/20$.