8. Compound Angles

Algebraic sum of two or more angles is called a compound angle. If \(A, B, C\) are any angle then \(A + B, A - B, A - B + C, A + B + C, A - B - C, A + B -C\) etc. are all compound angles.

8.1. The Addition Formula

\(\sin(A + B) = \sin A\cos B + \sin B\cos A\)

\(\cos(A + B) = \cos A\cos B - \sin A\sin B\)

\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}\)

compound angle addition

Consider the diagram above. \(PM\) and \(PN\) are perpendicualr to \(OQ\) and \(ON\). \(RN\) is parallel to \(OQ\) and \(NQ\) is perpendicular to \(OQ\). The left diagram represents the case when sum of angles is an acute angle while the right diagram represents the case when sum of angles is an obtuse angle.

\(\angle RPN = 90^{\circ} - \angle PNR = \angle RNO = \angle NOQ = \angle A\)

Now we can write, \(\sin(A + B) = \sin QOP = \frac{MP}{OP} = \frac{MR + RP}{OP} = \frac{QN}{OP} + \frac{RP}{OP}\)

\(=\frac{QN}{ON}\frac{ON}{OP} + \frac{RP}{NP}\frac{NP}{OP} = \sin A\cos B + \cos A\sin B\)

Also, \(\cos(A + B) = \cos QOP = \frac{OM}{OP} = \frac{OQ - MQ}{OP} = \frac{OQ}{ON}\frac{ON}{OP} - \frac{RN}{NP}\frac{NP}{OP}\)

\(= \cos A\cos B - \sin A\sin B\)

These two results lead to \(\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}\)

We have shown that addition formula is true when angles involved are acute angles. The same proof can be applied to prove the results for all values of \(A\) and \(B\).

Consider \(A' = 90^{\circ} + A \therefore \) and \(\cos A' = \sin A\)

\(\sin(A' + B) = \cos (A + B) = \cos A\cos B - \sin A\sin B = \sin A'\cos B + \cos A'\sin B\)

Similarly \(\cos(A' + B) = -\sin(A + B) = -\sin A\cos B - \sin B\cos A = \cos A'\cos B - \sin A'\sin B\)

We can prove it again for \(B' = 90^{\circ} + B\) and so on by increasing the values of \(A\) and \(B\). Then we can again increase values by \(90^{\circ}\) and proceeding this way we see that the formula holds true for all values of \(A\) and \(B\).

8.2. The Subtraction Formula

\(\sin(A - B) = \sin A\cos B - \sin B\cos A\)

\(\cos(A - B) = \cos A\cos B + \sin A\sin B\)

\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}\)

compound angle subtraction

Conside the diagram above. The angle \(MOP\) is \(A - B.\) We take a point \(P,\) and draw \(PM\) and \(PN\) perpendicular to \(OM\) and \(ON\) respectively. From \(N\) we draw \(NQ\) and \(NR\) perpendicular to \(OQ\) and \(MP\) respectively.

\(\angle RPN = 90^{\circ} - \angle PNR = \angle QON = A\)

Thus, we can write \(\sin(A - B) = \sin MOP = \frac{MP}{OP} = \frac{MR - PR}{OP} = \frac{QN}{ON}\frac{ON}{OP} - \frac{PR}{PN}\frac{PN}{OP}\)

Thus, \(\sin(A - B) = \sin A\cos B - \cos A\sin B\)

Also, \(\cos(A - B) = \frac{OM}{OP} = \frac{OQ + QM}{OP} = \frac{OQ}{ON}\frac{ON}{OP} + \frac{RN}{NP}\frac{NP}{OP}\)

\(= \cos A\cos B + \sin A\sin B\)

We have shown that subtraction formula is true when angles involved are acute angles. The same proof can be applied to prove the results for all values of \(A\) and \(B\).

From the results obtained we find upon division that \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}\)

8.3. Important Deductions

  1. \(\sin(A + B)\sin(A - B) = \sin^2A - \sin^2B = \cos^2B - \cos^2A\)

    Proof: L.H.S. \(= (\sin A\cos B + \sin B\cos A)(\sin A\cos B - \sin B\cos A)\)

    \(= \sin^2A\cos^2B - \sin^2B\cos^2A = \sin^2A(1 - \sin^2B) - \sin^2B(1 - \sin^2A)\)

    \(= \sin^2A - \sin^2A\sin^2B - \sin^2B + \sin^2B\sin^2A\)

    \(= =\sin^2A - \sin^2B = (1 - \cos^2A) - (1 - \cos^2B)\)

    \(= \cos^2B - \cos^2A\)

  2. \(\cos(A + B)\cos(A - B) = \cos^2A - \sin^2B = \cos^2B - \sin^2A\)

    Proof: L.H.S. \(=(\cos A\cos B - \sin A\sin B)(\cos A\cos B + \sin A\sin B)\)

    \(= \cos^2A\cos^2B - \sin^2A\sin^2B = \cos^2A(1- \sin^2B) - (1 - \cos^2A)\sin^2B\)

    \(=\cos^2A - \cos^2A\sin^2B - \sin^2B + \cos^2A\sin^2B = \cos^2A - \sin^2B = \cos^2B - \sin^2A\)

  3. \(\cot(A + B) = \frac{\cot A\cot B - 1}{\cot B + \cot A}\)

    Proof: L.H.S. \(= \cot(A + B) = \frac{\cos(A + B)}{\sin(A + B)}\)

    \(= \frac{\cos A\cos B - \sin A\sin B}{\sin A\cos B + \cos A\sin B}\)

    Dividing numberator and denominator by \(\sin A\sin B\)

    \(= \frac{\cot A\cot B - 1}{\cot B + \cot A}\)

  4. \(\cot(A - B) = \frac{\cot A\cot B + 1}{\cot B - \cot A}\)

    Proof: L.H.S. \(= \cot(A - B) = \frac{\cos(A - B)}{\sin(A - B)}\)

    \(= \frac{\cos A\cos B + \sin A\sin B}{\sin A\cos B - \cos A\sin B}\)

    Dividing numberator and denominator by \(\sin A\sin B\)

    \(= \frac{\cot A\cot B + 1}{\cot B - \cot A}\)

  5. \(\tan(A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}\)

    Proof: L.H.S. \(= \tan[(A + B) + C] = \frac{\tan(A + B) + \tan C}{1 - \tan(A + B)\tan C}\)

    \(= \frac{\frac{\tan A + \tan B}{1 - \tan A\tan B} + \tan C}{1 - \frac{\tan A + \tan B}{1 - \tan A\tan B}\tan C}\)

    \(= \frac{\frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B}}{\frac{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}{1 - \tan A\tan B}}\)

    \(= \frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}\)

8.4. To express \(a\cos\theta + b\sin\theta\) in the form of \(k\cos\phi\) or \(k\sin\phi\)

\(a\cos\theta + b\sin\theta = \sqrt{a^2 + b^2}\left(\frac{a}{\sqrt{a^2 + b^2}}\cos\theta + \frac{b}{\sqrt{a^2 + b^2}}\sin\theta\right)\)

Let \(\cos\alpha = \frac{a}{\sqrt{a^2 + b^2}}\) then \(\sin\alpha = \frac{b}{\sqrt{a^2 + b^2}}\)

Thus, \(a\cos\theta + b\sin\theta = \sqrt{a^2 + b^2}(\cos\alpha\cos\theta + \sin\alpha\sin\theta)\)

\(= \sqrt{a^2 + b^2}\cos(\theta - \alpha) = k\cos\phi\) where \(k = \sqrt{a^2 + b^2}\) and \(\phi = \theta - \alpha\)

Alternatively, if \(\frac{a}{\sqrt{a^2 + b^2}} = \sin\alpha\) then \(\frac{b}{\sqrt{a^2 + b^2}} = \cos\alpha\)

Thus, \(a\cos\theta + b\sin\theta = \sqrt{a^2 + b^2}(\sin\alpha\cos\theta + \cos\alpha + \sin\theta)\)

\(= \sqrt{a^2 + b^2}\sin(\theta + \alpha) = k\sin\phi\) where \(k = \sqrt{a^2+b^2}\) and \(\phi = \theta + \alpha\)

8.5. Problems

  1. If \(\sin\alpha = \frac{3}{5}\) and \(\cos\beta = \frac{9}{41},\) find the values of \(\sin(\alpha - \beta)\) and \(\cos(\alpha + \beta).\)

  2. If \(\sin\alpha = \frac{45}{53}\) and \(\sin\beta = \frac{33}{65},\) find the values of \(\sin(\alpha - \beta)\) and \(\sin(\alpha + \beta).\)

  3. If \(\sin\alpha = \frac{15}{17}\) and \(\cos\beta = \frac{12}{13},\) find the values of \(\sin(\alpha + \beta), \cos(\alpha - \beta)\) and \(\tan(\alpha + beta).\)

Prove the following:

  1. \(\cos(45^{\circ} - A)\cos(45^{\circ} - B) - \sin(45^{\circ} - A)\sin(45^{\circ} - B) = \sin(A + B)\)

  2. \(\sin(45^{\circ} + A)\cos(45^\circ - B) + \cos(45^{\circ} + A)\sin(45^\circ - B) = \cos(A - B).\)

  3. \(\frac{\sin(A - B)}{\cos A\cos B} + \frac{\sin(B - C)}{\cos B\cos C} + \frac{\sin(C - A)}{\cos C\cos A} = 0\)

  4. \(\sin 105^\circ + \cos 105^\circ = \cos 45^\circ\)

  5. \(\sin 75^\circ - \sin 15^\circ = \cos 105^\circ + \cos 15^\circ\)

  6. \(\cos\alpha\cos(\gamma - \alpha) - \sin\alpha\sin(\gamma - \alpha) = \cos\gamma\)

  7. \(\cos(\alpha + \beta)\cos\gamma - \cos(\beta + \gamma)\cos\alpha = \sin\beta\sin(\gamma - \alpha)\)

  8. \(\sin(n + 1)A\sin(n - 1)A + \cos(n + 1)A\cos(n - 1)A = \cos 2A\)

  9. \(\sin(n + 1)A\sin(n + 2)A + \cos(n + 1)A\cos(n + 2)A = \cos A\)

  10. Find the value of \(\cos 15^\circ\) and \(\sin 105^\circ\)

  11. Find the value of \(\tan 105^\circ\)

  12. Find the value of \(\frac{\tan 495^\circ}{\cot 855^\circ}\)

  13. Evaluate \(\sin\left(n\pi + (-1)^n \frac{\pi}{4}\right),\) where \(n\) is an integer.

Prove the following:

  1. \(\sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}}\)

  2. \(\cos 75^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}}\)

  3. \(\tan 75^\circ = 2 + \sqrt{3}\)

  4. \(\tan 15^\circ = 2 - \sqrt{3}\)

Find the value of following:

  1. \(\cos 1395^\circ\)

  2. \(\tan(-330^\circ)\)

  3. \(\sin 300^\circ \cosec 1050^\circ - \tan(-120^\circ)\)

  4. \(\tan\left(\frac{11\pi}{12}\right)\)

  5. \(\tan \left((-1)^n\frac{\pi}{4}\right)\)

Prove the following:

  1. \(\cos 18^\circ - \sin 18^\circ = \sqrt{2}\sin 27^\circ\)

  2. \(\tan 70^\circ = 2\tan 50^\circ + \tan 20^\circ\)

  3. \(\cot\left(\frac{\pi}{4} + x\right)\cot\left(\frac{\pi}{4} - x\right) = 1\)

  4. \(\cos(m + n)\theta.\cos(m - n)\theta - \sin(m + n)\theta\sin(m - n)\theta = \cos 2m\theta\)

  5. \(\frac{\tan(\theta + \phi) + \tan(\theta - \phi)}{1 - \tan(\theta + \phi)\tan(\theta - \phi)} = \tan 2\theta\)

  6. \(\cos 9^\circ + \sin 9^\circ = \sqrt{2}\sin 54^\circ\)

  7. \(\frac{\cos 20^\circ - \sin 20^\circ}{\cos 20^\circ + \sin 20^\circ} = \tan 25^\circ\)

  8. \(\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin(A + B)}{\sin(A - B)}\)

  9. \(\frac{1}{\tan 3A - \tan A} - \frac{1}{\cot 3A - \cot A} = \cot 2A\)

  10. \(\frac{1}{\tan 3A + \tan A} - \frac{1}{\cot 3A - \cot A} = \cot 4A\)

  11. \(\frac{\sin 3\alpha}{\sin\alpha} + \frac{\cos 3\alpha}{cos\alpha} = 4\cos 2\alpha\)

  12. \(\frac{\tan\left(\frac{\pi}{4} + A \right) - \tan\left(\frac{\pi}{4} - A\right)}{\tan\left(\frac{\pi}{4} + A\right) + \tan\left(\frac{\pi}{4} - A\right)} = \sin 2A\)

  13. \(\tan 40^\circ + 2 \tan 10^\circ = \tan 50^\circ\)

  14. \(\tan(\alpha + \beta)\tan(\alpha - \beta) = \frac{\sin^2\alpha - \sin^2\beta}{\cos^2\alpha - \sin^2\beta}\)

  15. \(\tan^2\alpha -\tan^2\beta = \frac{\sin(\alpha + \beta)\sin(\alpha - \beta)}{\cos^2\alpha\cos^2\beta}\)

  16. \(\tan[(2n + 1)\pi + \theta] + \tan[(2n + 1)\pi - \theta] = 0\)

  17. \(\tan\left(\frac{\pi}{4} + \theta\right)\tan\left(\frac{3\pi}{4} + \theta\right) + 1 = 0\)

  18. If \(\tan\alpha = p\) and \(\tan\beta = q\) prove that \(\cos(\alpha + \beta) = \frac{1 - pq}{\sqrt{(1 + p^2)(1 + q^2)}}\)

  19. if \(\tan \beta = \frac{2\sin\alpha\sin\gamma}{\sin(\alpha + \gamma)},\) show that \(\cot\alpha, \cot\beta, \cot\gamma\) are in A.P.

  20. Eliminate \(\theta\) if \(\tan(\theta - \alpha) = a\) and \(\tan(\theta + \alpha) = b\)

  21. Eliminate \(\alpha\) and \(\beta\) if \(\tan\alpha + \tan\beta = b, \cot\alpha + \cot\beta = a\) and \(\alpha + \beta = \gamma\)

  22. If \(A + B = 45^\circ,\) show that \((1 + \tan A)(1 + \tan B) = 2\)

  23. If \(\sin\alpha\sin\beta - \cos\alpha\cos\beta + 1 = 0,\) prove that \(1 + \cot\alpha\tan\beta = 0\)

  24. If \(\tan\beta = \frac{n\sin\alpha\cos\alpha}{1 - n\sin^2\alpha},\) prove that \(\tan(\alpha - \beta) = (1 - n)\alpha\)

  25. If \(\cos(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta) = -\frac{3}{2},\) prove that \(\cos\alpha + \cos\beta + \cos\gamma = \sin\alpha + \sin\beta + \sin\gamma = 0\)

  26. If \(\tan\alpha = \frac{m}{m + 1}, \tan\beta = \frac{1}{2m + 1},\) prove that \(\alpha + \beta = \frac{\pi}{4}\)

  27. If \(A + B = 45^\circ,\) show that \((\cot A - 1)(\cot B - 1) = 2\)

  28. If \(\tan\alpha - \tan\beta = x\) and \(\cot\beta - \cot\alpha = y,\) prove that \(\cot(\alpha - \beta) = \frac{x + y}{xy}\)

  29. If a right angle be divided into three pats \(\alpha, \beta\) and \(\gamma,\) prove that \(\cot\alpha = \frac{\tan\beta + \tan\gamma}{1 - \tan\beta\tan\gamma}\)

  30. If \(2\tan\beta + \cot \beta = \tan\alpha,\) show that \(\cot \beta = 2\tan(\alpha - \beta)\)

  31. If in any \(\triangle ABC, C = 90^\circ,\) prove that \(\cosec(A - B) = \frac{a^2 + b^2}{a^2 - b^2}\) and \(\sec(A - B) = \frac{c^2}{2ab}\)

  32. If \(\cot A = \sqrt{ac}, \cot B = \sqrt{\frac{c}{a}}, \tan C = \sqrt{\frac{c}{a^3}}\) and \(c = a^2 + a + 1,\) prove that \(A = B + C\)

  33. If \(\frac{\tan(A - B)}{\tan A} + \frac{\sin^2C}{\sin^2A} = 1,\) prove that \(\tan A\tan B = \tan^2 C\)

  34. If \(\sin\alpha\sin\beta - \cos\alpha\cos\beta = 1\) show that \(\tan\alpha + \tan\beta = 0\)

  35. If \(\sin\theta = 3\sin(\theta + 2\alpha),\) prove that \(\tan(\theta + \alpha),\) prove that \(\tan(\theta + \alpha) + 2\tan\alpha = 0\)

  36. If \(3\tan\theta\tan\phi = 1,\) prove that \(2\cos(\theta + \phi) = \cos(\theta - \alpha)\)

  37. Find the sign of the expression \(\sin\theta + \cos\theta\) when \(\theta = 100^\circ\)

  38. Prove that the value of \(5\cos\theta + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3\) lies between \(-4\) and \(10\)

  39. If \(m\tan(\theta - 30^\circ) = n\tan(\theta + 120^\circ),\) show that \(\cos2\theta = \frac{m + n}{2(m - n)}\)

  40. if \(\alpha + \beta = \theta\) and \(\tan\alpha:\tan\beta = x:y,\) prove that \(\sin(\alpha - \beta) = \frac{x - y}{x + y}\sin\theta\)

  41. Find the maximum and minimum value of \(7\cos\theta + 24\sin\theta\)

  42. Show that \(\sin 100^\circ - \sin 10^\circ\) is positive.