Mathematics · Textbook solutions

Trigonometry - II

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 166 questions

3.1 Compound Angles

8 q

Solved Examples

Worked · 8
  1. 3.1 SolvedEx.1
    Find the value of cos⁡15∘\cos 15^\circ.
  2. 3.1 SolvedEx.2
    Find the value of tan⁡13π12\tan \frac{13\pi}{12}.
  3. 3.1 SolvedEx.3
    Show that sin⁡(x+y)sin⁡(x−y)=tan⁡x+tan⁡ytan⁡x−tan⁡y\frac{\sin(x+y)}{\sin(x-y)} = \frac{\tan x + \tan y}{\tan x - \tan y}.
  4. 3.1 SolvedEx.4
    Show that : tan⁡3x tan⁡2x tan⁡x=tan⁡3x−tan⁡2x−tan⁡x\tan 3x \, \tan 2x \, \tan x = \tan 3x - \tan 2x - \tan x
  5. 3.1 SolvedEx.5
    Show that cos⁡(π4+x)+cos⁡(π4−x)=2 cos⁡x\cos\left(\frac{\pi}{4} + x\right) + \cos\left(\frac{\pi}{4} - x\right) = \sqrt{2}\,\cos x.
  6. 3.1 SolvedEx.6
    If tan⁡A−tan⁡B=x\tan A - \tan B = x and cot⁡B−cot⁡A=y\cot B - \cot A = y then show that cot⁡(A−B)=1x+1y\cot(A-B) = \frac{1}{x} + \frac{1}{y}.
  7. 3.1 SolvedEx.7
    If tan⁡α=1x(x2+x+1)\tan\alpha = \frac{1}{\sqrt{x(x^2 + x + 1)}}, tan⁡β=xx2+x+1\tan\beta = \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}} and tan⁡γ=x−3+x−2+x−1\tan\gamma = \sqrt{x^{-3} + x^{-2} + x^{-1}}, then show that α+β=γ\alpha + \beta = \gamma.
  8. 3.1 SolvedEx.8
    If sin⁡A+sin⁡B=x\sin A + \sin B = x and cos⁡A+cos⁡B=y\cos A + \cos B = y then show that sin⁡(A+B)=2xyx2+y2\sin(A+B) = \frac{2xy}{x^2 + y^2}.

Exercise 3.1

22 q
  1. Find the values of
    Ex 3.1 Q1(i)
    sin⁡15∘\sin 15^\circ
  2. Ex 3.1 Q1(ii)
    cos⁡75∘\cos 75^\circ
  3. Ex 3.1 Q1(iii)
    tan⁡105∘\tan 105^\circ
  4. Ex 3.1 Q1(iv)
    cot⁡225∘\cot 225^\circ
  5. Prove the following.
    Ex 3.1 Q2(i)
    cos⁡(π2−x)cos⁡(π2−y)−sin⁡(π2−x)sin⁡(π2−y)=−cos⁡(x+y)\cos\left(\frac{\pi}{2} - x\right)\cos\left(\frac{\pi}{2} - y\right) - \sin\left(\frac{\pi}{2} - x\right)\sin\left(\frac{\pi}{2} - y\right) = -\cos(x+y)
  6. Ex 3.1 Q2(ii)
    tan⁡(π4+θ)=1−tan⁡θ1+tan⁡θ\tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 - \tan\theta}{1 + \tan\theta}
  7. Ex 3.1 Q2(iii)
    (1+tan⁡x1−tan⁡x)2=tan⁡(π4+x)tan⁡(π4−x)\left(\frac{1 + \tan x}{1 - \tan x}\right)^{2} = \frac{\tan\left(\frac{\pi}{4} + x\right)}{\tan\left(\frac{\pi}{4} - x\right)}
  8. Ex 3.1 Q2(iv)
    sin⁡[(n+1)A]⋅sin⁡[(n+2)A]+cos⁡[(n+1)A]⋅cos⁡[(n+2)A]=cos⁡A\sin[(n+1)A]\cdot\sin[(n+2)A] + \cos[(n+1)A]\cdot\cos[(n+2)A] = \cos A
  9. Ex 3.1 Q2(v)
    2 cos⁡(π4−A)=cos⁡A+sin⁡A\sqrt{2}\,\cos\left(\frac{\pi}{4} - A\right) = \cos A + \sin A
  10. Ex 3.1 Q2(vi)
    cos⁡(x−y)cos⁡(x+y)=cot⁡xcot⁡y+1cot⁡xcot⁡y−1\frac{\cos(x - y)}{\cos(x + y)} = \frac{\cot x \cot y + 1}{\cot x \cot y - 1}
  11. Ex 3.1 Q2(vii)
    cos⁡(x+y)⋅cos⁡(x−y)=cos⁡2y−sin⁡2x\cos(x+y)\cdot\cos(x-y) = \cos^{2}y - \sin^{2}x
  12. Ex 3.1 Q2(viii)
    tan⁡5A−tan⁡3Atan⁡5A+tan⁡3A=sin⁡2Asin⁡8A\frac{\tan 5A - \tan 3A}{\tan 5A + \tan 3A} = \frac{\sin 2A}{\sin 8A}
  13. Ex 3.1 Q2(ix)
    tan⁡8θ−tan⁡5θ−tan⁡3θ=tan⁡8θtan⁡5θtan⁡3θ\tan 8\theta - \tan 5\theta - \tan 3\theta = \tan 8\theta \tan 5\theta \tan 3\theta
  14. Ex 3.1 Q2(x)
    tan⁡50∘=tan⁡40∘+2tan⁡10∘\tan 50^\circ = \tan 40^\circ + 2\tan 10^\circ
  15. Ex 3.1 Q2(xi)
    cos⁡27∘+sin⁡27∘cos⁡27∘−sin⁡27∘=tan⁡72∘\frac{\cos 27^\circ + \sin 27^\circ}{\cos 27^\circ - \sin 27^\circ} = \tan 72^\circ
  16. Ex 3.1 Q2(xii)
    tan⁡10∘+tan⁡35∘+tan⁡10∘⋅tan⁡35∘=1\tan 10^\circ + \tan 35^\circ + \tan 10^\circ \cdot \tan 35^\circ = 1
  17. Ex 3.1 Q2(xiii)
    cot⁡Acot⁡4A+1cot⁡Acot⁡4A−1=cos⁡3Acos⁡5A\frac{\cot A \cot 4A + 1}{\cot A \cot 4A - 1} = \frac{\cos 3A}{\cos 5A}
  18. Ex 3.1 Q2(xiv)
    cos⁡15∘−sin⁡15∘cos⁡15∘+sin⁡15∘=13\frac{\cos 15^\circ - \sin 15^\circ}{\cos 15^\circ + \sin 15^\circ} = \frac{1}{\sqrt{3}}
  19. If sin⁡A=−513\sin A = \frac{-5}{13}, π<A<3π2\pi < A < \frac{3\pi}{2} and cos⁡B=35\cos B = \frac{3}{5}, 3π2<B<2π\frac{3\pi}{2} < B < 2\pi then find
    Ex 3.1 Q3(i)
    sin⁡(A+B)\sin(A+B)
  20. Ex 3.1 Q3(ii)
    cos⁡(A−B)\cos(A-B)
  21. Ex 3.1 Q3(iii)
    tan⁡(A+B)\tan(A+B)
  22. Ex 3.1 Q4
    If tan⁡A=56\tan A = \frac{5}{6}, tan⁡B=111\tan B = \frac{1}{11}, prove that A+B=π4A + B = \frac{\pi}{4}.

3.2 Allied Angles

11 q

Solved Examples

Worked · 11
  1. Find the values of
    3.2 SolvedEx.1(i)
    (sin⁡495∘)(\sin 495^\circ)
  2. 3.2 SolvedEx.1(ii)
    cos⁡930∘\cos 930^\circ
  3. 3.2 SolvedEx.1(iii)
    tan⁡840∘\tan 840^\circ
  4. Show that :
    3.2 SolvedEx.2(i)
    cos⁡24∘+cos⁡55∘+cos⁡125∘+cos⁡204∘+cos⁡300∘=12\cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2}
  5. 3.2 SolvedEx.2(ii)
    sec⁡840∘⋅cot⁡(−945∘)+sin⁡600∘⋅tan⁡(−690∘)=32\sec 840^\circ \cdot \cot(-945^\circ) + \sin 600^\circ \cdot \tan(-690^\circ) = \frac{3}{2}
  6. 3.2 SolvedEx.2(iii)
    cosec⁡(90∘−θ)⋅sin⁡(180∘−θ)cot⁡(360∘−θ)sec⁡(180∘+θ)tan⁡(90∘+θ)sin⁡(−θ)=1\frac{\operatorname{cosec}(90^\circ - \theta) \cdot \sin(180^\circ - \theta)\cot(360^\circ - \theta)}{\sec(180^\circ + \theta)\tan(90^\circ + \theta)\sin(-\theta)} = 1
  7. 3.2 SolvedEx.2(iv)
    cot⁡(π2+θ)sin⁡(−θ)cot⁡(π−θ)cos⁡(2π−θ)sin⁡(π+θ)tan⁡(2π−θ)=−cosec⁡θ\frac{\cot\left(\frac{\pi}{2} + \theta\right)\sin(-\theta)\cot(\pi - \theta)}{\cos(2\pi - \theta)\sin(\pi + \theta)\tan(2\pi - \theta)} = -\operatorname{cosec}\theta
  8. Prove the following :
    3.2 SolvedEx.3(i)
    sin⁡π15+sin⁡4π15−sin⁡14π15−sin⁡11π15=0\sin\frac{\pi}{15} + \sin\frac{4\pi}{15} - \sin\frac{14\pi}{15} - \sin\frac{11\pi}{15} = 0
  9. 3.2 SolvedEx.3(ii)
    sin⁡2(π4−x)+sin⁡2(π4+x)=1\sin^{2}\left(\frac{\pi}{4} - x\right) + \sin^{2}\left(\frac{\pi}{4} + x\right) = 1
  10. 3.2 SolvedEx.3(iii)
    sin⁡2π8+sin⁡23π8+sin⁡25π8+sin⁡27π8=2\sin^{2}\frac{\pi}{8} + \sin^{2}\frac{3\pi}{8} + \sin^{2}\frac{5\pi}{8} + \sin^{2}\frac{7\pi}{8} = 2
  11. 3.2 SolvedEx.3(iv)
    cos⁡2(π10)+cos⁡2(2π5)+cos⁡2(3π5)+cos⁡2(9π10)=2\cos^{2}\left(\frac{\pi}{10}\right) + \cos^{2}\left(\frac{2\pi}{5}\right) + \cos^{2}\left(\frac{3\pi}{5}\right) + \cos^{2}\left(\frac{9\pi}{10}\right) = 2

Exercise 3.2

16 q
  1. Find the value of :
    Ex 3.2 Q1(i)
    sin⁡690∘\sin 690^\circ
  2. Ex 3.2 Q1(ii)
    sin⁡(495∘)\sin (495^\circ)
  3. Ex 3.2 Q1(iii)
    cos⁡315∘\cos 315^\circ
  4. Ex 3.2 Q1(iv)
    cos⁡(600∘)\cos (600^\circ)
  5. Ex 3.2 Q1(v)
    tan⁡225∘\tan 225^\circ
  6. Ex 3.2 Q1(vi)
    tan⁡(−690∘)\tan (-690^\circ)
  7. Ex 3.2 Q1(vii)
    sec⁡240∘\sec 240^\circ
  8. Ex 3.2 Q1(viii)
    sec⁡(−855∘)\sec (-855^\circ)
  9. Ex 3.2 Q1(ix)
    cosec⁡780∘\operatorname{cosec} 780^\circ
  10. Ex 3.2 Q1(x)
    cot⁡(−1110∘)\cot (-1110^\circ)
  11. Prove the following:
    Ex 3.2 Q2(i)
    cos⁡(π+x)cos⁡(−x)sin⁡(π−x)cos⁡(π2+x)=cot⁡2x\dfrac{\cos(\pi + x)\cos(-x)}{\sin(\pi - x)\cos\left(\dfrac{\pi}{2} + x\right)} = \cot^2 x
  12. Ex 3.2 Q2(ii)
    cos⁡(3π2+x)cos⁡(2π+x)[cot⁡(3π2−x)+cot⁡(2π+x)]=1\cos\left(\dfrac{3\pi}{2} + x\right)\cos(2\pi + x)\left[\cot\left(\dfrac{3\pi}{2} - x\right) + \cot(2\pi + x)\right] = 1
  13. Ex 3.2 Q2(iii)
    sec⁡840∘⋅cot⁡(−945∘)+sin⁡600∘tan⁡(−690∘)=32\sec 840^\circ \cdot \cot(-945^\circ) + \sin 600^\circ \tan(-690^\circ) = \dfrac{3}{2}
  14. Ex 3.2 Q2(iv)
    cosec⁡(90∘−x)sin⁡(180∘−x)cot⁡(360∘−x)sec⁡(180∘+x)tan⁡(90∘+x)sin⁡(−x)=1\dfrac{\operatorname{cosec}(90^\circ - x)\sin(180^\circ - x)\cot(360^\circ - x)}{\sec(180^\circ + x)\tan(90^\circ + x)\sin(-x)} = 1
  15. Ex 3.2 Q2(v)
    sin⁡3(π+x)sec⁡2(π−x)tan⁡(2π−x)cos⁡2(π2+x)sin⁡(π−x)cosec⁡2−x=tan⁡3x\dfrac{\sin^3(\pi + x)\sec^2(\pi - x)\tan(2\pi - x)}{\cos^2\left(\dfrac{\pi}{2} + x\right)\sin(\pi - x)\operatorname{cosec}^2 - x} = \tan^3 x
  16. Ex 3.2 Q2(vi)
    cos⁡θ+sin⁡(270∘+θ)−sin⁡(270∘−θ)+cos⁡(180∘+θ)=0\cos\theta + \sin(270^\circ + \theta) - \sin(270^\circ - \theta) + \cos(180^\circ + \theta) = 0

3.3 Multiple Angles — Double and Triple

12 q

Solved Examples

Worked · 12
  1. 3.3.2 SolvedEx.1
    Prove that 1+tan⁡θ tan⁡θ2=sec⁡θ1 + \tan\theta\,\tan\dfrac{\theta}{2} = \sec\theta.
  2. 3.3.2 SolvedEx.2
    Prove that tan⁡20∘ tan⁡40∘ tan⁡60∘ tan⁡80∘=3\tan 20^\circ\,\tan 40^\circ\,\tan 60^\circ\,\tan 80^\circ = 3.
  3. 3.3.2 SolvedEx.3
    Prove that 2cosec⁡2x+cosec⁡x=sec⁡x⋅cot⁡(x2)2\operatorname{cosec} 2x + \operatorname{cosec} x = \sec x\cdot\cot\left(\dfrac{x}{2}\right).
  4. 3.3.2 SolvedEx.4
    Prove that cos⁡3θ−cos⁡3θcos⁡θ+sin⁡3θ+sin⁡3θsin⁡θ=3\dfrac{\cos^3\theta - \cos 3\theta}{\cos\theta} + \dfrac{\sin^3\theta + \sin 3\theta}{\sin\theta} = 3.
  5. 3.3.2 SolvedEx.5
    Prove that tan⁡5A+tan⁡3Atan⁡5A−tan⁡3A=4cos⁡2A⋅cos⁡4A\dfrac{\tan 5A + \tan 3A}{\tan 5A - \tan 3A} = 4\cos^2 A\cdot\cos 4A.
  6. 3.3.2 SolvedEx.6
    Show that (cos⁡θ+isin⁡θ)3=cos⁡3θ+isin⁡3θ(\cos\theta + i\sin\theta)^3 = \cos 3\theta + i\sin 3\theta, where i2=−1i^2 = -1.
  7. 3.3.2 SolvedEx.7
    Show that 4sin⁡θcos⁡3θ−4cos⁡θsin⁡3θ=sin⁡4θ4\sin\theta\cos^3\theta - 4\cos\theta\sin^3\theta = \sin 4\theta.
  8. 3.3.2 SolvedEx.8
    Show that 1+sin⁡2A1−sin⁡2A=tan⁡(π4+A)\sqrt{\dfrac{1 + \sin 2A}{1 - \sin 2A}} = \tan\left(\dfrac{\pi}{4} + A\right).
  9. 3.3.2 SolvedEx.9
    Find sin⁡x2\sin\dfrac{x}{2}, cos⁡x2\cos\dfrac{x}{2}, tan⁡x2\tan\dfrac{x}{2} if tan⁡x=43\tan x = \dfrac{4}{3}, xx lies in II quadrant.
  10. 3.3.2 SolvedEx.10
    Find the value of tan⁡π8\tan\dfrac{\pi}{8}.
  11. 3.3.2 SolvedEx.11
    Prove that cos⁡2x+cos⁡2(x+π3)+cos⁡2(x−π3)=32\cos^2 x + \cos^2\left(x + \dfrac{\pi}{3}\right) + \cos^2\left(x - \dfrac{\pi}{3}\right) = \dfrac{3}{2}.
  12. 3.3.2 SolvedEx.12
    Find sin⁡π10\sin\dfrac{\pi}{10}.

Exercise 3.3

22 q
  1. Find values of :
    Ex 3.3 Q1(i)
    sin⁡π8\sin\frac{\pi}{8}
  2. Ex 3.3 Q1(ii)
    cos⁡π8\cos\frac{\pi}{8}
  3. Ex 3.3 Q2
    Find sin⁡2x\sin 2x, cos⁡2x\cos 2x, tan⁡2x\tan 2x if sec⁡x=−135\sec x = \frac{-13}{5}, π2<x<π\frac{\pi}{2} < x < \pi.
  4. Prove the following:
    Ex 3.3 Q3(i)
    1−cos⁡2θ1+cos⁡2θ=tan⁡2θ\frac{1-\cos 2\theta}{1+\cos 2\theta} = \tan^2\theta
  5. Ex 3.3 Q3(ii)
    (sin⁡3x+sin⁡x)sin⁡x+(cos⁡3x−cos⁡x)cos⁡x=0(\sin 3x + \sin x)\sin x + (\cos 3x - \cos x)\cos x = 0
  6. Ex 3.3 Q3(iii)
    (cos⁡x+cos⁡y)2+(sin⁡x−sin⁡y)2=4cos⁡2(x+y)2(\cos x + \cos y)^2 + (\sin x - \sin y)^2 = 4\cos^2\frac{(x+y)}{2}
  7. Ex 3.3 Q3(iv)
    (cos⁡x−cos⁡y)2+(sin⁡x−sin⁡y)2=4sin⁡2(x−y)2(\cos x - \cos y)^2 + (\sin x - \sin y)^2 = 4\sin^2\frac{(x-y)}{2}
  8. Ex 3.3 Q3(v)
    tan⁡x+cot⁡x=2cosec⁡2x\tan x + \cot x = 2\operatorname{cosec} 2x
  9. Ex 3.3 Q3(vi)
    cos⁡x+sin⁡xcos⁡x−sin⁡x−cos⁡x−sin⁡xcos⁡x+sin⁡x=2tan⁡2x\frac{\cos x + \sin x}{\cos x - \sin x} - \frac{\cos x - \sin x}{\cos x + \sin x} = 2\tan 2x
  10. Ex 3.3 Q3(vii)
    2+2+2+2cos⁡8x=2cos⁡x\sqrt{2+\sqrt{2+\sqrt{2+2\cos 8x}}} = 2\cos x
  11. Ex 3.3 Q3(viii)
    16sin⁡θcos⁡θcos⁡2θcos⁡4θcos⁡8θ=sin⁡16θ16\sin\theta\cos\theta\cos 2\theta\cos 4\theta\cos 8\theta = \sin 16\theta
  12. Ex 3.3 Q3(ix)
    sin⁡3xcos⁡x+cos⁡3xsin⁡x=2cot⁡2x\frac{\sin 3x}{\cos x} + \frac{\cos 3x}{\sin x} = 2\cot 2x
  13. Ex 3.3 Q3(x)
    cos⁡x1+sin⁡x=cot⁡(x2)−1cot⁡(x2)+1\frac{\cos x}{1+\sin x} = \frac{\cot\left(\frac{x}{2}\right)-1}{\cot\left(\frac{x}{2}\right)+1}
  14. Ex 3.3 Q3(xi)
    tan⁡(θ2)+cot⁡(θ2)cot⁡(θ2)−tan⁡(θ2)=sec⁡θ\frac{\tan\left(\frac{\theta}{2}\right)+\cot\left(\frac{\theta}{2}\right)}{\cot\left(\frac{\theta}{2}\right)-\tan\left(\frac{\theta}{2}\right)} = \sec\theta
  15. Ex 3.3 Q3(xii)
    1tan⁡3A−tan⁡A−1cot⁡3A−cot⁡A=cot⁡2A\frac{1}{\tan 3A - \tan A} - \frac{1}{\cot 3A - \cot A} = \cot 2A
  16. Ex 3.3 Q3(xiii)
    cos⁡7∘cos⁡14∘cos⁡28∘cos⁡56∘=sin⁡68∘16cos⁡83∘\cos 7^\circ\cos 14^\circ\cos 28^\circ\cos 56^\circ = \frac{\sin 68^\circ}{16\cos 83^\circ}
  17. Ex 3.3 Q3(xiv)
    sin⁡2(−160∘)sin⁡270∘+sin⁡(180∘−θ)sin⁡θ=sec⁡220∘\frac{\sin^2(-160^\circ)}{\sin^2 70^\circ} + \frac{\sin(180^\circ-\theta)}{\sin\theta} = \sec^2 20^\circ
  18. Ex 3.3 Q3(xv)
    2cos⁡4x+12cos⁡x+1=(2cos⁡x−1)(2cos⁡2x−1)\frac{2\cos 4x + 1}{2\cos x + 1} = (2\cos x - 1)(2\cos 2x - 1)
  19. Ex 3.3 Q3(xvi)
    cos⁡2x+cos⁡2(x+120∘)+cos⁡2(x−120∘)=32\cos^2 x + \cos^2(x+120^\circ) + \cos^2(x-120^\circ) = \frac{3}{2}
  20. Ex 3.3 Q3(xvii)
    2cosec⁡2x+cosec⁡x=sec⁡xcot⁡(x2)2\operatorname{cosec} 2x + \operatorname{cosec} x = \sec x\cot\left(\frac{x}{2}\right)
  21. Ex 3.3 Q3(xviii)
    4cos⁡xcos⁡(x+π3)+cos⁡2(π−π3)=cos⁡3x4\cos x\cos\left(x+\frac{\pi}{3}\right) + \cos^2\left(\pi-\frac{\pi}{3}\right) = \cos 3x
  22. Ex 3.3 Q3(xix)
    sin⁡xtan⁡(x2)+2cos⁡x=21+tan⁡2(x2)\sin x\tan\left(\frac{x}{2}\right) + 2\cos x = \frac{2}{1+\tan^2\left(\frac{x}{2}\right)}

3.4 Factorization and Defactorization Formulae

10 q

Solved Examples

Worked · 10
  1. Prove the following :
    3.4.2 SolvedEx.1(i)
    sin⁡40∘−cos⁡70∘=3cos⁡80∘\sin 40^\circ - \cos 70^\circ = \sqrt{3}\cos 80^\circ
  2. 3.4.2 SolvedEx.1(ii)
    cos⁡40∘+cos⁡50∘+cos⁡70∘+cos⁡80∘=cos⁡20∘+cos⁡10∘\cos 40^\circ + \cos 50^\circ + \cos 70^\circ + \cos 80^\circ = \cos 20^\circ + \cos 10^\circ
  3. Express the following as sum or difference of two trigonometric function:
    3.4.2 SolvedEx.2(i)
    2sin⁡4θcos⁡2θ2\sin 4\theta\cos 2\theta
  4. 3.4.2 SolvedEx.2(ii)
    4sin⁡(A+B2)sin⁡(A−B2)4\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)
  5. Show that
    3.4.2 SolvedEx.3(i)
    sin⁡8x+sin⁡2xcos⁡2x−cos⁡8x=cos⁡3x\frac{\sin 8x + \sin 2x}{\cos 2x - \cos 8x} = \cos 3x
  6. 3.4.2 SolvedEx.3(ii)
    sin⁡2α+sin⁡2βsin⁡2α−sin⁡2β=tan⁡(α+β)tan⁡(α−β)\frac{\sin 2\alpha + \sin 2\beta}{\sin 2\alpha - \sin 2\beta} = \frac{\tan(\alpha+\beta)}{\tan(\alpha-\beta)}
  7. Prove that following.
    3.4.2 SolvedEx.4(i)
    cos⁡(7x−5y)+cos⁡(7y−5x)sin⁡(7x−5y)+sin⁡(7y−5x)=cot⁡(x+y)\frac{\cos(7x-5y)+\cos(7y-5x)}{\sin(7x-5y)+\sin(7y-5x)} = \cot(x+y)
  8. 3.4.2 SolvedEx.4(ii)
    sin⁡6θ+sin⁡4θ−sin⁡2θ=4cos⁡θsin⁡2θcos⁡3θ\sin 6\theta + \sin 4\theta - \sin 2\theta = 4\cos\theta\sin 2\theta\cos 3\theta
  9. 3.4.2 SolvedEx.4(iii)
    cos⁡3xsin⁡9x−sin⁡xcos⁡5xcos⁡xcos⁡5x−sin⁡3xsin⁡9x=tan⁡8x\frac{\cos 3x\sin 9x - \sin x\cos 5x}{\cos x\cos 5x - \sin 3x\sin 9x} = \tan 8x
  10. 3.4.2 SolvedEx.4(iv)
    cos⁡20∘cos⁡40∘cos⁡60∘cos⁡80∘=116\cos 20^\circ\cos 40^\circ\cos 60^\circ\cos 80^\circ = \frac{1}{16}

Exercise 3.4

10 q
  1. Express the following as a sum or difference of two trigonometric function.
    Ex 3.4 Q1(i)
    2sin⁡4x cos⁡2x2\sin 4x\,\cos 2x
  2. Ex 3.4 Q1(ii)
    2sin⁡2π3 cos⁡π22\sin \dfrac{2\pi}{3}\,\cos \dfrac{\pi}{2}
  3. Ex 3.4 Q1(iii)
    2cos⁡4θ cos⁡2θ2\cos 4\theta\,\cos 2\theta
  4. Ex 3.4 Q1(iv)
    2cos⁡35∘ cos⁡75∘2\cos 35^\circ\,\cos 75^\circ
  5. Prove the following :
    Ex 3.4 Q2(i)
    sin⁡2x+sin⁡2ysin⁡2x−sin⁡2y=tan⁡(x+y)tan⁡(x−y)\dfrac{\sin 2x + \sin 2y}{\sin 2x - \sin 2y} = \dfrac{\tan(x + y)}{\tan(x - y)}
  6. Ex 3.4 Q2(ii)
    sin⁡6x+sin⁡4x−sin⁡2x=4cos⁡x sin⁡2x cos⁡3x\sin 6x + \sin 4x - \sin 2x = 4\cos x\,\sin 2x\,\cos 3x
  7. Ex 3.4 Q2(iii)
    sin⁡x−sin⁡3x+sin⁡5x−sin⁡7xcos⁡x−cos⁡3x−cos⁡5x+cos⁡7x=cot⁡2x\dfrac{\sin x - \sin 3x + \sin 5x - \sin 7x}{\cos x - \cos 3x - \cos 5x + \cos 7x} = \cot 2x
  8. Ex 3.4 Q2(iv)
    sin⁡18∘ cos⁡39∘+sin⁡6∘ cos⁡15∘=sin⁡24∘ cos⁡33∘\sin 18^\circ\,\cos 39^\circ + \sin 6^\circ\,\cos 15^\circ = \sin 24^\circ\,\cos 33^\circ
  9. Ex 3.4 Q2(v)
    cos⁡20∘ cos⁡40∘ cos⁡60∘ cos⁡80∘=116\cos 20^\circ\,\cos 40^\circ\,\cos 60^\circ\,\cos 80^\circ = \dfrac{1}{16}
  10. Ex 3.4 Q2(vi)
    sin⁡20∘ sin⁡40∘ sin⁡60∘ sin⁡80∘=316\sin 20^\circ\,\sin 40^\circ\,\sin 60^\circ\,\sin 80^\circ = \dfrac{3}{16}

3.5 Trigonometric Functions of Angles of a Triangle

6 q

Solved Examples

Worked · 6
  1. In ΔABC\Delta ABC prove that
    3.5 SolvedEx.1(i)
    sin⁡2A+sin⁡2B−sin⁡2C=4cos⁡A cos⁡B sin⁡C\sin 2A + \sin 2B - \sin 2C = 4\cos A\,\cos B\,\sin C
  2. 3.5 SolvedEx.1(ii)
    cos⁡A+cos⁡B+cos⁡C=1+4sin⁡A2 sin⁡B2 sin⁡C2\cos A + \cos B + \cos C = 1 + 4\sin\dfrac{A}{2}\,\sin\dfrac{B}{2}\,\sin\dfrac{C}{2}
  3. 3.5 SolvedEx.1(iii)
    sin⁡2A+sin⁡2B−sin⁡2C=2sin⁡A sin⁡B cos⁡C\sin^2 A + \sin^2 B - \sin^2 C = 2\sin A\,\sin B\,\cos C
  4. 3.5 SolvedEx.1(iv)
    cot⁡A cot⁡B+cot⁡B cot⁡+cot⁡C cot⁡A=1\cot A\,\cot B + \cot B\,\cot + \cot C\,\cot A = 1
  5. 3.5 SolvedEx.1(v)
    tan⁡A2 tan⁡B2+tan⁡B2 tan⁡C2+tan⁡C2 tan⁡A2=1\tan\dfrac{A}{2}\,\tan\dfrac{B}{2} + \tan\dfrac{B}{2}\,\tan\dfrac{C}{2} + \tan\dfrac{C}{2}\,\tan\dfrac{A}{2} = 1
  6. 3.5 SolvedEx.1(vi)
    cos⁡A−cos⁡B+cos⁡C+1cos⁡A+cos⁡B+cos⁡C−1=cot⁡A2 cot⁡C2\dfrac{\cos A - \cos B + \cos C + 1}{\cos A + \cos B + \cos C - 1} = \cot\dfrac{A}{2}\,\cot\dfrac{C}{2}

Exercise 3.5

8 q
  1. Ex 3.5 Q1
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that cos⁡2A+cos⁡2B+cos⁡2C=−1−4cos⁡Acos⁡Bcos⁡C\cos 2A + \cos 2B + \cos 2C = -1 - 4\cos A\cos B\cos C.
  2. Ex 3.5 Q2
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that sin⁡A+sin⁡B+sin⁡C=4cos⁡A2cos⁡B2cos⁡C2\sin A + \sin B + \sin C = 4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\cos\dfrac{C}{2}.
  3. Ex 3.5 Q3
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that cos⁡A+cos⁡B−cos⁡C=4cos⁡A2cos⁡B2sin⁡C2−1\cos A + \cos B - \cos C = 4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\sin\dfrac{C}{2} - 1.
  4. Ex 3.5 Q4
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that sin⁡2A+sin⁡2B+sin⁡2C=2+2cos⁡Acos⁡Bcos⁡C\sin^{2} A + \sin^{2} B + \sin^{2} C = 2 + 2\cos A\cos B\cos C.
  5. Ex 3.5 Q5
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that sin⁡2A2+sin⁡2B2−sin⁡2C2=1−2cos⁡A2cos⁡B2sin⁡C2\sin^{2}\dfrac{A}{2} + \sin^{2}\dfrac{B}{2} - \sin^{2}\dfrac{C}{2} = 1 - 2\cos\dfrac{A}{2}\cos\dfrac{B}{2}\sin\dfrac{C}{2}.
  6. Ex 3.5 Q6
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that cot⁡A2+cot⁡B2+cot⁡C2=cot⁡A2cot⁡B2cot⁡C2\cot\dfrac{A}{2} + \cot\dfrac{B}{2} + \cot\dfrac{C}{2} = \cot\dfrac{A}{2}\cot\dfrac{B}{2}\cot\dfrac{C}{2}.
  7. Ex 3.5 Q7
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that tan⁡2A+tan⁡2B+tan⁡2C=tan⁡2A tan⁡2B tan⁡2C\tan 2A + \tan 2B + \tan 2C = \tan 2A\,\tan 2B\,\tan 2C.
  8. Ex 3.5 Q8
    In ΔABC\Delta ABC, A+B+C=πA + B + C = \pi. Show that cos⁡2A+cos⁡2B−cos⁡2C=1−2sin⁡Asin⁡Bcos⁡C\cos^{2} A + \cos^{2} B - \cos^{2} C = 1 - 2\sin A\sin B\cos C.

Miscellaneous Exercise 3

41 q

(I) Select the correct option

Practice · 10
  1. Misc I Q1
    The value of sin⁡(n+1)A sin⁡(n+2)A+cos⁡(n+1)A cos⁡(n+2)A\sin (n+1)A\,\sin (n+2)A + \cos (n+1)A\,\cos (n+2)A is equal to
    1. A.
      sin⁡A\sin A
    2. B.
      cos⁡A\cos A
    3. C.
      −cos⁡A-\cos A
    4. D.
      sin⁡2A\sin 2A
  2. Misc I Q2
    If tan⁡A−tan⁡B=x\tan A - \tan B = x and cot⁡B−cot⁡A=y\cot B - \cot A = y then cot⁡(A−B)=\cot (A - B) = …\ldots
    1. A.
      1y−1x\dfrac{1}{y} - \dfrac{1}{x}
    2. B.
      1x−1y\dfrac{1}{x} - \dfrac{1}{y}
    3. C.
      1x+1y\dfrac{1}{x} + \dfrac{1}{y}
    4. D.
      xyx−y\dfrac{xy}{x-y}
  3. Misc I Q3
    If sin⁡θ=nsin⁡(θ+2α)\sin\theta = n\sin (\theta + 2\alpha) then tan⁡(θ+α)\tan (\theta + \alpha) is equal to
    1. A.
      1+n2−n tan⁡α\dfrac{1+n}{2-n}\,\tan\alpha
    2. B.
      1−n1+n tan⁡α\dfrac{1-n}{1+n}\,\tan\alpha
    3. C.
      tan⁡α\tan\alpha
    4. D.
      1+n1−n tan⁡α\dfrac{1+n}{1-n}\,\tan\alpha
  4. Misc I Q4
    The value of cos⁡θ1+sin⁡θ\dfrac{\cos\theta}{1 + \sin\theta} is equal to……\ldots\ldots
    1. A.
      tan⁡(θ2−π4)\tan\left(\dfrac{\theta}{2} - \dfrac{\pi}{4}\right)
    2. B.
      tan⁡(−π4−θ2)\tan\left(-\dfrac{\pi}{4} - \dfrac{\theta}{2}\right)
    3. C.
      tan⁡(π4−θ2)\tan\left(\dfrac{\pi}{4} - \dfrac{\theta}{2}\right)
    4. D.
      tan⁡(π4+θ2)\tan\left(\dfrac{\pi}{4} + \dfrac{\theta}{2}\right)
  5. Misc I Q5
    The value of cos⁡A cos⁡(60∘−A) cos⁡(60∘+A)\cos A\,\cos (60^\circ - A)\,\cos (60^\circ + A) is equal to……\ldots\ldots
    1. A.
      12cos⁡3A\dfrac{1}{2}\cos 3A
    2. B.
      cos⁡3A\cos 3A
    3. C.
      14cos⁡3A\dfrac{1}{4}\cos 3A
    4. D.
      4cos⁡3A4\cos 3A
  6. Misc I Q6
    The value of sin⁡π14 sin⁡3π14 sin⁡5π14 sin⁡7π14 sin⁡9π14 sin⁡11π14 sin⁡13π14\sin\dfrac{\pi}{14}\,\sin\dfrac{3\pi}{14}\,\sin\dfrac{5\pi}{14}\,\sin\dfrac{7\pi}{14}\,\sin\dfrac{9\pi}{14}\,\sin\dfrac{11\pi}{14}\,\sin\dfrac{13\pi}{14} is ….\ldots.
    1. A.
      116\dfrac{1}{16}
    2. B.
      164\dfrac{1}{64}
    3. C.
      1128\dfrac{1}{128}
    4. D.
      1256\dfrac{1}{256}
  7. Misc I Q7
    If α+β+ϰ=π\alpha + \beta + \varkappa = \pi then the value of sin⁡2α+sin⁡2β−sin⁡2ϰ\sin^{2}\alpha + \sin^{2}\beta - \sin^{2}\varkappa is equal to……\ldots\ldots
    1. A.
      2sin⁡α2\sin\alpha
    2. B.
      2sin⁡α cos⁡β sin⁡ϰ2\sin\alpha\,\cos\beta\,\sin\varkappa
    3. C.
      2sin⁡α sin⁡β cos⁡ϰ2\sin\alpha\,\sin\beta\,\cos\varkappa
    4. D.
      2sin⁡α sin⁡β sin⁡ϰ2\sin\alpha\,\sin\beta\,\sin\varkappa
  8. Misc I Q8
    Let 0<A, B<π20 < A,\ B < \dfrac{\pi}{2} satisfying the equation 3sin⁡2A+2sin⁡2B=13\sin^{2} A + 2\sin^{2} B = 1 and 3sin⁡2A−2sin⁡2B=03\sin 2A - 2\sin 2B = 0 then A+2BA + 2B is equal to….\ldots.
    1. A.
      π\pi
    2. B.
      π2\dfrac{\pi}{2}
    3. C.
      π4\dfrac{\pi}{4}
    4. D.
      2π2\pi
  9. Misc I Q9
    In ΔABC\Delta ABC if cot⁡A cot⁡B cot⁡C>0\cot A\,\cot B\,\cot C > 0 then the triangle is….\ldots.
    1. A.
      Acute angled
    2. B.
      right angled
    3. C.
      obtuse angled
    4. D.
      isosceles right angled
  10. Misc I Q10
    The numerical value of tan⁡20∘ tan⁡80∘ cot⁡50∘\tan 20^\circ\,\tan 80^\circ\,\cot 50^\circ is equal to……\ldots\ldots
    1. A.
      3\sqrt{3}
    2. B.
      13\dfrac{1}{\sqrt{3}}
    3. C.
      232\sqrt{3}
    4. D.
      123\dfrac{1}{2\sqrt{3}}

(II) Prove the following

Practice · 31
  1. Misc II Q1
    Prove that tan⁡20∘ tan⁡80∘ cot⁡50∘=3\tan 20^\circ\,\tan 80^\circ\,\cot 50^\circ = \sqrt{3}.
  2. Misc II Q2
    If sin⁡α sin⁡β−cos⁡α cos⁡β+1=0\sin\alpha\,\sin\beta - \cos\alpha\,\cos\beta + 1 = 0 then prove that cot⁡α tan⁡β=−1\cot\alpha\,\tan\beta = -1.
  3. Misc II Q3
    Prove that cos⁡2π15 cos⁡4π15 cos⁡8π15 cos⁡16π15=116\cos\dfrac{2\pi}{15}\,\cos\dfrac{4\pi}{15}\,\cos\dfrac{8\pi}{15}\,\cos\dfrac{16\pi}{15} = \dfrac{1}{16}.
  4. Misc II Q4
    Prove that (1+cos⁡π8)(1+cos⁡3π8)(1+cos⁡5π8)(1+cos⁡7π8)=18\left(1 + \cos\dfrac{\pi}{8}\right)\left(1 + \cos\dfrac{3\pi}{8}\right)\left(1 + \cos\dfrac{5\pi}{8}\right)\left(1 + \cos\dfrac{7\pi}{8}\right) = \dfrac{1}{8}.
  5. Misc II Q5
    Prove that cos⁡12∘+cos⁡84∘+cos⁡156∘+cos⁡132∘=−12\cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ = -\dfrac{1}{2}.
  6. Misc II Q6
    Prove that cos⁡(π4+x)+cos⁡(π4−x)=2 cos⁡x\cos\left(\dfrac{\pi}{4} + x\right) + \cos\left(\dfrac{\pi}{4} - x\right) = \sqrt{2}\,\cos x.
  7. Misc II Q7
    Prove that sin⁡5x−2sin⁡3x+sin⁡xcos⁡5x−cos⁡x=tan⁡x\dfrac{\sin 5x - 2\sin 3x + \sin x}{\cos 5x - \cos x} = \tan x.
  8. Misc II Q8
    Prove that sin⁡26x−sin⁡24x=sin⁡2x sin⁡10x\sin^{2} 6x - \sin^{2} 4x = \sin 2x\,\sin 10x.
  9. Misc II Q9
    Prove that cos⁡22x−cos⁡26x=sin⁡4x sin⁡8x\cos^{2} 2x - \cos^{2} 6x = \sin 4x\,\sin 8x.
  10. Misc II Q10
    Prove that cot⁡4x (sin⁡5x+sin⁡3x)=cot⁡x (sin⁡5x−sin⁡3x)\cot 4x\,(\sin 5x + \sin 3x) = \cot x\,(\sin 5x - \sin 3x).
  11. Misc II Q11
    Prove that cos⁡9x−cos⁡5xsin⁡17x−sin⁡3x=− sin⁡2xcos⁡10x\dfrac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\,\dfrac{\sin 2x}{\cos 10x}.
  12. Misc II Q12
    If sin⁡2A=λsin⁡2B\sin 2A = \lambda\sin 2B then prove that tan⁡(A+B)tan⁡(A−B)=λ+1λ−1\dfrac{\tan (A + B)}{\tan (A - B)} = \dfrac{\lambda + 1}{\lambda - 1}.
  13. Misc II Q13
    Prove that 2cos⁡2A+12cos⁡2A−1=tan⁡(60∘+A) tan⁡(60∘−A)\dfrac{2\cos 2A + 1}{2\cos 2A - 1} = \tan (60^\circ + A)\,\tan (60^\circ - A).
  14. Misc II Q14
    Prove that tan⁡A+tan⁡(60∘+A)+tan⁡(120∘+A)=3tan⁡3A\tan A + \tan (60^\circ + A) + \tan (120^\circ + A) = 3\tan 3A.
  15. Misc II Q15
    Prove that 3tan⁡610∘−27tan⁡410∘+33tan⁡210∘=13\tan^{6}10^\circ - 27\tan^{4}10^\circ + 33\tan^{2}10^\circ = 1.
  16. Misc II Q16
    Prove that cosec⁡48∘+cosec⁡96∘+cosec⁡192∘+cosec⁡384∘=0\operatorname{cosec} 48^\circ + \operatorname{cosec} 96^\circ + \operatorname{cosec} 192^\circ + \operatorname{cosec} 384^\circ = 0.
  17. Misc II Q17
    Prove that 3(sin⁡x−cos⁡x)4+6(sin⁡x+cos⁡x)2+4(sin⁡6x+cos⁡6x)=133(\sin x - \cos x)^{4} + 6(\sin x + \cos x)^{2} + 4(\sin^{6} x + \cos^{6} x) = 13.
  18. Misc II Q18
    Prove that tan⁡A+2tan⁡2A+4tan⁡4A+8cot⁡8A=cot⁡A\tan A + 2\tan 2A + 4\tan 4A + 8\cot 8A = \cot A.
  19. Misc II Q19
    If A+B+C=3π2A + B + C = \dfrac{3\pi}{2} then prove that cos⁡2A+cos⁡2B+cos⁡2C=1−4sin⁡A sin⁡B sin⁡C\cos 2A + \cos 2B + \cos 2C = 1 - 4\sin A\,\sin B\,\sin C.
  20. Misc II Q20
    In any triangle ABCABC, sin⁡A−cos⁡B=cos⁡C\sin A - \cos B = \cos C then prove that ∠B=π2\angle B = \dfrac{\pi}{2}.
  21. Misc II Q21
    Prove that tan⁡3x1+tan⁡2x+cot⁡3x1+cot⁡2x=sec⁡x cosec⁡x−2sin⁡x cos⁡x\dfrac{\tan^{3} x}{1 + \tan^{2} x} + \dfrac{\cot^{3} x}{1 + \cot^{2} x} = \sec x\,\operatorname{cosec} x - 2\sin x\,\cos x.
  22. Misc II Q22
    Prove that sin⁡20∘ sin⁡40∘ sin⁡80∘=38\sin 20^\circ\,\sin 40^\circ\,\sin 80^\circ = \dfrac{\sqrt{3}}{8}.
  23. Misc II Q23
    Prove that sin⁡18∘=5−14\sin 18^\circ = \dfrac{\sqrt{5} - 1}{4}.
  24. Misc II Q24
    Prove that cos⁡36∘=5+14\cos 36^\circ = \dfrac{\sqrt{5} + 1}{4}.
  25. Misc II Q25
    Prove that sin⁡36∘=10−254\sin 36^\circ = \dfrac{\sqrt{10 - 2\sqrt{5}}}{4}.
  26. Misc II Q26
    Prove that sin⁡πc8=122−2\sin\dfrac{\pi^{c}}{8} = \dfrac{1}{2}\sqrt{2 - \sqrt{2}}.
  27. Misc II Q27
    Prove that tan⁡π8=2−1\tan\dfrac{\pi}{8} = \sqrt{2} - 1.
  28. Misc II Q28
    Prove that tan⁡6∘ tan⁡42∘ tan⁡66∘ tan⁡78∘=1\tan 6^\circ\,\tan 42^\circ\,\tan 66^\circ\,\tan 78^\circ = 1.
  29. Misc II Q29
    Prove that sin⁡47∘+sin⁡61∘−sin⁡11∘−sin⁡25∘=cos⁡7∘\sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = \cos 7^\circ.
  30. Misc II Q30
    Prove that 3 cosec⁡20∘−sec⁡20∘=4\sqrt{3}\,\operatorname{cosec} 20^\circ - \sec 20^\circ = 4.
  31. Misc II Q31
    In ΔABC\Delta ABC, ∠C=2π3\angle C = \dfrac{2\pi}{3} then prove that cos⁡2A+cos⁡2B−cos⁡A cos⁡B=34\cos^{2} A + \cos^{2} B - \cos A\,\cos B = \dfrac{3}{4}.