Practice section 3 of 8

Express $\dfrac{x^2+6x+9}{(x-2)^2(x+1)}$ in partial fractions.

Express $\dfrac{2x^2+x-3}{(x+1)(2x^2+3x+4)}$ in partial fractions.

Express $\dfrac{3x^3+x-4}{(3x+1)(x^2+1)}$ in partial fractions.

In triangle $ABC$, $AB=1$, $BC=3$ and angle $ABC=\theta$ radians. Show that $AC=\sqrt{10-6\cos\theta}$.

Find the principal value of (a) $\sin^{-1}(1/2)$; (b) $\cos^{-1}(1/\sqrt2)$; (c) $\tan^{-1}(1/\sqrt3)$; (d) $\sin[\cos^{-1}(-\sqrt3/2)]$.

If $\theta=\cos^{-1}(-1/\sqrt3)$, where $0<\theta<\pi$, find the exact values of (i) $\sin\theta$ and (ii) $\sin2\theta$.