Mathematics Pure Math 2,3 9709: TRIGONOMETRY Topical Past Paper Questions

(a) Use the substitution $ x = 2 \sin u $ to show that $$ \int_{0}^{1}\frac{3x^{3}}{\sqrt{4-x^{2}}}\mathrm{d}x=\int_{a}^{b}24\sin^{3}u\mathrm{d}u, $$ where $a$ and $b$ are exact values to be found. (b) Hence, find the exact value of $ \int_{0}^{1}\frac{3x^{3}}{\sqrt{4-x^{2}}}dx $…

(a) Express $3\sqrt{3}\sin\left(\theta+\frac{1}{6}\pi\right)-2\sin\theta$ in the form $R\sin(\theta+\alpha)$, where $R>0$ and $0 [5] (b) Hence, solve the equation $3\sqrt{3}\sin\left(2x+\frac{1}{6}\pi\right)-2\sin2x=\sqrt{6}$ for $0 [4]

(a) Show that $\cos^4\theta - \sin^4\theta - 4\sin^2\theta\cos^2\theta \equiv \cos^2 2\theta + \cos 2\theta - 1$. [3] (b) Solve the equation $\cos^4\alpha - \sin^4\alpha = 4\sin^2\alpha\cos^2\alpha$ for $0^\circ \leqslant \alpha \leqslant 180^\circ$. [3]