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) \(\cos2y-\cos2x\)