Exercises 20 (Compound and multiple angles)

1. Without using a calculator find:

   1. $\sin 15^\circ$
   2. $\cos 15^\circ$
   3. $\tan 15^\circ$
   4. $\sin 105^\circ$
   5. $\cos 105^\circ$

2. If $\cos A = 0.6$, $\cos B = 0.8$, and that both $A$ and $B$ are acute angles, determine without using a calculator the values of $\sin (A + B)$ and $\cos(A + B)$.

3. Given that $\cos A = 0.2$, $\cos B = 0.5$, and that both $A$ and $B$ are acute angles, find the values of $\sin (A + B)$ and $\cos (A - B)$ (without using a calculator to determine $A$ and $B$).

4. Prove that $\sin(\theta+45^\circ)=\frac{1}{\sqrt{2}}(\sin\theta+\cos\theta)$.

5. If $\tan(A+B)=0.75$ and $\tan A=2$, determine $B$.

6. Show that $\sin (\theta + 90^\circ) + \cos (\theta - 180^\circ) = 0$.

7. Determine the acute angle $\theta$, which satisfies the equation $3 \cos (\theta - 14^\circ) = 4 \sin \theta$.

8. Prove that:

   1. $\dfrac{\cos2\theta}{\cos\theta+\sin\theta}=\cos\theta-\sin\theta$
   2. $\dfrac{1-\cos2\theta}{1+\cos2\theta}=\tan^2\theta$

9. Show that $\sin 3A = 3\sin A - 4\sin^3 A$.

10. Express as a sum or difference:

    1. $2\cos 5\theta \cos 3\theta$
    2. $\sin 25^\circ \cos 65^\circ$
    3. $\cos 3x \cos x$
    4. $\sin 5x \cos x$
    5. $2\sin (x + y) \sin (x - y)$

11. Show that $\sin(\theta+45^\circ)\sin(45^\circ-\theta) = -\frac{1}{2}\cos2\theta$.

12. Prove that $\dfrac{\sin A+\sin 3A + \sin 5A}{\cos A+\cos 3A+\cos 5A}=\tan3A$.

13. Prove that $\dfrac{\sin A-\sin 5A +\sin 9A -\sin13A}{\cos A-\cos 5A-\cos 9A+\cos 13 A}=\cot 4A$.

14. (* difficult) Show that in any triangle $\tan\frac{B-C}{2}=\frac{b-c}{b+c}\cot\frac{A}{2}$.

Solutions: 1. (i) $\frac{\sqrt{3}-1}{2\sqrt{2}}$; (ii) $\frac{\sqrt{3}+1}{2\sqrt{2}}$; (iii) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$; (iv) $\frac{\sqrt{3}+1}{2\sqrt{2}}$; (v) $\frac{1-\sqrt{3}}{2\sqrt{2}}$; 2. $\sin(A+B)=1$, $\cos(A+B)=0$; 3. $\sin(A+B)=\sqrt{6}/5+\sqrt{3}/10\approx 0.6631$ (4 d.p.), $\cos(A-B)=1/10+3\sqrt{2}/5\approx 0.9465$ (4 d.p.); 5. $B=-26.6^\circ$; 7. $\theta= 41.6^\circ$; 10. (i) $\cos8\theta+\cos2\theta$; (ii) $\frac12(1-\sin40^\circ)$; (iii) $\frac12(\cos 4x+\cos2x)$; (iv) $\frac12(\sin6x+\sin4x)$; (v) $\cos2y-\cos2x$