1.52: Compound-angle identities, equations and integration

4PM1/2/November/2020 — Question 10 · 16 marks

(a) Show that

\mathrm{cos}\,(A+B)+\mathrm{cos}\,(A-B)=2\mathrm{cos}\,A\,\mathrm{cos}\,B.

(2)

(b) Hence show that

\mathrm{cos}\,P+\mathrm{cos}\,Q =2\mathrm{cos}\left(\frac{P+Q}{2}\right) \mathrm{cos}\left(\frac{P-Q}{2}\right).

(3)

0\leq\theta\leq\frac{\pi}{2}

, the equation

\mathrm{cos}\,5\theta+\mathrm{cos}\,7\theta=0.

Give each solution in terms of

\pi

(4)

(d) Show that

\mathrm{cos}\,8x+2\mathrm{cos}\,6x+\mathrm{cos}\,4x =4\mathrm{cos}\,6x\,\mathrm{cos}^2x.

(3)

(e) Use calculus to find the exact value of

\int_0^{\frac{\pi}{3}}\mathrm{cos}\,6x\,\mathrm{cos}^2x\,\mathrm{d}x.

(4)