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Mathematics · Textbook solutions

Trigonometric Functions

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. · 83 questions

3.2 Angles and their Measurement

20 q

Solved Examples

Worked · 5
  1. 3.1 Eg.1
    Convert 40∘ 20′40^\circ\,20' into radian measure.
  2. 3.1 Eg.2
    Convert 6 radians into degree measure.
  3. 3.1 Eg.3
    Find the radius of the circle in which a central angle of 60∘60^\circ intercepts an arc of length 37.4 cm (use π=227\pi = \frac{22}{7}).
  4. 3.1 Eg.4
    The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use π=3.14\pi = 3.14).
  5. 3.1 Eg.5
    If the arcs of the same lengths in two circles subtend angles 65∘65^\circ and 110∘110^\circ at the centre, find the ratio of their radii.

Exercise 3.1

Practice · 15
  1. Find the radian measures corresponding to the following degree measures:
    Ex 3.1 Q1(i)
    25∘25^\circ
  2. Ex 3.1 Q1(ii)
    −47∘30′-47^\circ 30'
  3. Ex 3.1 Q1(iii)
    240∘240^\circ
  4. Ex 3.1 Q1(iv)
    520∘520^\circ
  5. Find the degree measures corresponding to the following radian measures (Use π=227\pi = \frac{22}{7}).
    Ex 3.1 Q2(i)
    1116\frac{11}{16}
  6. Ex 3.1 Q2(ii)
    −4-4
  7. Ex 3.1 Q2(iii)
    5π3\frac{5\pi}{3}
  8. Ex 3.1 Q2(iv)
    7π6\frac{7\pi}{6}
  9. Ex 3.1 Q3
    A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
  10. Ex 3.1 Q4
    Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π=227\pi = \frac{22}{7}).
  11. Ex 3.1 Q5
    In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
  12. Ex 3.1 Q6
    If in two circles, arcs of the same length subtend angles 60∘60^\circ and 75∘75^\circ at the centre, find the ratio of their radii.
  13. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
    Ex 3.1 Q7(i)
    10 cm
  14. Ex 3.1 Q7(ii)
    15 cm
  15. Ex 3.1 Q7(iii)
    21 cm

3.3 Trigonometric Functions

14 q

Solved Examples

Worked · 4
  1. 3.2 Eg.6
    If cos⁡x=−35\cos x = -\frac{3}{5}, xx lies in the third quadrant, find the values of other five trigonometric functions.
  2. 3.2 Eg.7
    If cot⁡x=−512\cot x = -\frac{5}{12}, xx lies in second quadrant, find the values of other five trigonometric functions.
  3. 3.2 Eg.8
    Find the value of sin⁡31π3\sin\frac{31\pi}{3}.
  4. 3.2 Eg.9
    Find the value of cos⁡(−1710∘)\cos(-1710^\circ).

Exercise 3.2

Practice · 10
  1. Ex 3.2 Q1
    Find the values of the other five trigonometric functions if cos⁡x=−12\cos x = -\frac{1}{2}, xx lies in third quadrant.
  2. Ex 3.2 Q2
    Find the values of the other five trigonometric functions if sin⁡x=35\sin x = \frac{3}{5}, xx lies in second quadrant.
  3. Ex 3.2 Q3
    Find the values of the other five trigonometric functions if cot⁡x=34\cot x = \frac{3}{4}, xx lies in third quadrant.
  4. Ex 3.2 Q4
    Find the values of the other five trigonometric functions if sec⁡x=135\sec x = \frac{13}{5}, xx lies in fourth quadrant.
  5. Ex 3.2 Q5
    Find the values of the other five trigonometric functions if tan⁡x=−512\tan x = -\frac{5}{12}, xx lies in second quadrant.
  6. Ex 3.2 Q6
    Find the value of the trigonometric function sin⁡765∘\sin 765^\circ.
  7. Ex 3.2 Q7
    Find the value of the trigonometric function cosec⁡(−1410∘)\operatorname{cosec}(-1410^\circ).
  8. Ex 3.2 Q8
    Find the value of the trigonometric function tan⁡19π3\tan\frac{19\pi}{3}.
  9. Ex 3.2 Q9
    Find the value of the trigonometric function sin⁡(−11π3)\sin\left(-\frac{11\pi}{3}\right).
  10. Ex 3.2 Q10
    Find the value of the trigonometric function cot⁡(−15π4)\cot\left(-\frac{15\pi}{4}\right).

3.4 Trigonometric Functions of Sum and Difference of Two Angles

34 q

Solved Examples

Worked · 8
  1. 3.3 Eg.10
    Prove that 3sin⁡π6sec⁡π3−4sin⁡5π6cot⁡π4=13\sin\frac{\pi}{6}\sec\frac{\pi}{3} - 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} = 1.
  2. 3.3 Eg.11
    Find the value of sin⁡15∘\sin 15^\circ.
  3. 3.3 Eg.12
    Find the value of tan⁡13π12\tan\frac{13\pi}{12}.
  4. 3.3 Eg.13
    Prove 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}.
  5. 3.3 Eg.14
    Show that tan⁡3xtan⁡2xtan⁡x=tan⁡3x−tan⁡2x−tan⁡x\tan 3x \tan 2x \tan x = \tan 3x - \tan 2x - \tan x.
  6. 3.3 Eg.15
    Prove that cos⁡(π4+x)+cos⁡(π4−x)=2cos⁡x\cos\left(\frac{\pi}{4} + x\right) + \cos\left(\frac{\pi}{4} - x\right) = \sqrt{2}\cos x.
  7. 3.3 Eg.16
    Prove that cos⁡7x+cos⁡5xsin⁡7x−sin⁡5x=cot⁡x\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \cot x.
  8. 3.3 Eg.17
    Prove that =sin⁡5x−2sin⁡3x+sin⁡xcos⁡5x−cos⁡x=tan⁡x= \frac{\sin 5x - 2\sin 3x + \sin x}{\cos 5x - \cos x} = \tan x

Exercise 3.3

Practice · 26
  1. Ex 3.3 Q1
    Prove that: sin⁡2π6+cos⁡2π3−tan⁡2π4=−12\sin^2 \frac{\pi}{6} + \cos^2 \frac{\pi}{3} - \tan^2 \frac{\pi}{4} = -\frac{1}{2}
  2. Ex 3.3 Q2
    Prove that: 2sin⁡2π6+cosec⁡27π6cos⁡2π3=322\sin^2 \frac{\pi}{6} + \operatorname{cosec}^2 \frac{7\pi}{6} \cos^2 \frac{\pi}{3} = \frac{3}{2}
  3. Ex 3.3 Q3
    Prove that: cot⁡2π6+cosec⁡5π6+3tan⁡2π6=6\cot^2 \frac{\pi}{6} + \operatorname{cosec} \frac{5\pi}{6} + 3\tan^2 \frac{\pi}{6} = 6
  4. Ex 3.3 Q4
    Prove that: 2sin⁡23π4+2cos⁡2π4+2sec⁡2π3=102\sin^2 \frac{3\pi}{4} + 2\cos^2 \frac{\pi}{4} + 2\sec^2 \frac{\pi}{3} = 10
  5. Find the value of:
    Ex 3.3 Q5(i)
    sin⁡75∘\sin 75^\circ
  6. Ex 3.3 Q5(ii)
    tan⁡15∘\tan 15^\circ
  7. Ex 3.3 Q6
    Prove the following: cos⁡(π4−x)cos⁡(π4−y)−sin⁡(π4−x)sin⁡(π4−y)=sin⁡(x+y)\cos\left(\frac{\pi}{4} - x\right)\cos\left(\frac{\pi}{4} - y\right) - \sin\left(\frac{\pi}{4} - x\right)\sin\left(\frac{\pi}{4} - y\right) = \sin(x + y)
  8. Ex 3.3 Q7
    Prove the following: tan⁡(π4+x)tan⁡(π4−x)=(1+tan⁡x1−tan⁡x)2\dfrac{\tan\left(\frac{\pi}{4} + x\right)}{\tan\left(\frac{\pi}{4} - x\right)} = \left(\dfrac{1 + \tan x}{1 - \tan x}\right)^2
  9. Ex 3.3 Q8
    Prove the following: cos⁡(π+x) cos⁡(−x)sin⁡(π−x) cos⁡(π2+x)=cot⁡2x\dfrac{\cos(\pi + x)\,\cos(-x)}{\sin(\pi - x)\,\cos\left(\frac{\pi}{2} + x\right)} = \cot^2 x
  10. Ex 3.3 Q9
    Prove the following: cos⁡(3π2+x)cos⁡(2π+x)[cot⁡(3π2−x)+cot⁡(2π+x)]=1\cos\left(\frac{3\pi}{2} + x\right)\cos(2\pi + x)\left[\cot\left(\frac{3\pi}{2} - x\right) + \cot(2\pi + x)\right] = 1
  11. Ex 3.3 Q10
    Prove the following: sin⁡(n+1)x sin⁡(n+2)x+cos⁡(n+1)x cos⁡(n+2)x=cos⁡x\sin(n + 1)x\,\sin(n + 2)x + \cos(n + 1)x\,\cos(n + 2)x = \cos x
  12. Ex 3.3 Q11
    Prove the following: cos⁡(3π4+x)−cos⁡(3π4−x)=−2 sin⁡x\cos\left(\frac{3\pi}{4} + x\right) - \cos\left(\frac{3\pi}{4} - x\right) = -\sqrt{2}\,\sin x
  13. Ex 3.3 Q12
    Prove the following: sin⁡26x−sin⁡24x=sin⁡2xsin⁡10x\sin^2 6x - \sin^2 4x = \sin 2x \sin 10x
  14. Ex 3.3 Q13
    Prove the following: cos⁡22x−cos⁡26x=sin⁡4xsin⁡8x\cos^2 2x - \cos^2 6x = \sin 4x \sin 8x
  15. Ex 3.3 Q14
    Prove the following: sin⁡2x+2sin⁡4x+sin⁡6x=4cos⁡2xsin⁡4x\sin 2x + 2\sin 4x + \sin 6x = 4\cos^2 x \sin 4x
  16. Ex 3.3 Q15
    Prove the following: cot⁡4x (sin⁡5x+sin⁡3x)=cot⁡x (sin⁡5x−sin⁡3x)\cot 4x\,(\sin 5x + \sin 3x) = \cot x\,(\sin 5x - \sin 3x)
  17. Ex 3.3 Q16
    Prove the following: cos⁡9x−cos⁡5xsin⁡17x−sin⁡3x=−sin⁡2xcos⁡10x\dfrac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\dfrac{\sin 2x}{\cos 10x}
  18. Ex 3.3 Q17
    Prove the following: sin⁡5x+sin⁡3xcos⁡5x+cos⁡3x=tan⁡4x\dfrac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x
  19. Ex 3.3 Q18
    Prove the following: sin⁡x−sin⁡ycos⁡x+cos⁡y=tan⁡x−y2\dfrac{\sin x - \sin y}{\cos x + \cos y} = \tan \dfrac{x - y}{2}
  20. Ex 3.3 Q19
    Prove the following: sin⁡x+sin⁡3xcos⁡x+cos⁡3x=tan⁡2x\dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x
  21. Ex 3.3 Q20
    Prove the following: sin⁡x−sin⁡3xsin⁡2x−cos⁡2x=2sin⁡x\dfrac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2\sin x
  22. Ex 3.3 Q21
    Prove the following: cos⁡4x+cos⁡3x+cos⁡2xsin⁡4x+sin⁡3x+sin⁡2x=cot⁡3x\dfrac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x
  23. Ex 3.3 Q22
    Prove the following: cot⁡xcot⁡2x−cot⁡2xcot⁡3x−cot⁡3xcot⁡x=1\cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1
  24. Ex 3.3 Q23
    Prove the following: tan⁡4x=4tan⁡x (1−tan⁡2x)1−6tan⁡2x+tan⁡4x\tan 4x = \dfrac{4\tan x\,(1 - \tan^2 x)}{1 - 6\tan^2 x + \tan^4 x}
  25. Ex 3.3 Q24
    Prove the following: cos⁡4x=1−8sin⁡2xcos⁡2x\cos 4x = 1 - 8\sin^2 x \cos^2 x
  26. Ex 3.3 Q25
    Prove the following: cos⁡6x=32cos⁡6x−48cos⁡4x+18cos⁡2x−1\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1

Miscellaneous Exercise on Chapter 3

15 q

Miscellaneous Examples

Worked · 5
  1. Misc Eg.18
    If sin⁡x=35\sin x = \frac{3}{5}, cos⁡y=−1213\cos y = -\frac{12}{13}, where xx and yy both lie in second quadrant, find the value of sin⁡(x+y)\sin(x + y).
  2. Misc Eg.19
    Prove that cos⁡2xcos⁡x2−cos⁡3xcos⁡9x2=sin⁡5xsin⁡5x2\cos 2x \cos\frac{x}{2} - \cos 3x \cos\frac{9x}{2} = \sin 5x \sin\frac{5x}{2}.
  3. Misc Eg.20
    Find the value of tan⁡π8\tan\frac{\pi}{8}.
  4. Misc Eg.21
    If tan⁡x=34\tan x = \frac{3}{4}, π<x<3π2\pi < x < \frac{3\pi}{2}, find the value of sin⁡x2\sin\frac{x}{2}, cos⁡x2\cos\frac{x}{2} and tan⁡x2\tan\frac{x}{2}.
  5. Misc Eg.22
    Prove that cos⁡2x+cos⁡2(x+π3)+cos⁡2(x−π3)=32\cos^2 x + \cos^2\left(x + \frac{\pi}{3}\right) + \cos^2\left(x - \frac{\pi}{3}\right) = \frac{3}{2}.

Miscellaneous Exercise

Practice · 10
  1. Misc Q1
    Prove that: 2cos⁡π13cos⁡9π13+cos⁡3π13+cos⁡5π13=02\cos\frac{\pi}{13}\cos\frac{9\pi}{13} + \cos\frac{3\pi}{13} + \cos\frac{5\pi}{13} = 0
  2. Misc Q2
    Prove that: (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
  3. Misc Q3
    Prove that: (cos⁡x+cos⁡y)2+(sin⁡x−sin⁡y)2=4cos⁡2x+y2(\cos x + \cos y)^2 + (\sin x - \sin y)^2 = 4\cos^2\frac{x + y}{2}
  4. Misc Q4
    Prove that: (cos⁡x−cos⁡y)2+(sin⁡x−sin⁡y)2=4sin⁡2x−y2(\cos x - \cos y)^2 + (\sin x - \sin y)^2 = 4\sin^2\frac{x - y}{2}
  5. Misc Q5
    Prove that: sin⁡x+sin⁡3x+sin⁡5x+sin⁡7x=4cos⁡xcos⁡2xsin⁡4x\sin x + \sin 3x + \sin 5x + \sin 7x = 4\cos x \cos 2x \sin 4x
  6. Misc Q6
    Prove that: (sin⁡7x+sin⁡5x)+(sin⁡9x+sin⁡3x)(cos⁡7x+cos⁡5x)+(cos⁡9x+cos⁡3x)=tan⁡6x\dfrac{(\sin 7x + \sin 5x) + (\sin 9x + \sin 3x)}{(\cos 7x + \cos 5x) + (\cos 9x + \cos 3x)} = \tan 6x
  7. Misc Q7
    Prove that: sin⁡3x+sin⁡2x−sin⁡x=4sin⁡xcos⁡x2cos⁡3x2\sin 3x + \sin 2x - \sin x = 4\sin x \cos\frac{x}{2}\cos\frac{3x}{2}
  8. Misc Q8
    Find sin⁡x2\sin\frac{x}{2}, cos⁡x2\cos\frac{x}{2} and tan⁡x2\tan\frac{x}{2}, given tan⁡x=−43\tan x = -\frac{4}{3}, xx in quadrant II.
  9. Misc Q9
    Find sin⁡x2\sin\frac{x}{2}, cos⁡x2\cos\frac{x}{2} and tan⁡x2\tan\frac{x}{2}, given cos⁡x=−13\cos x = -\frac{1}{3}, xx in quadrant III.
  10. Misc Q10
    Find sin⁡x2\sin\frac{x}{2}, cos⁡x2\cos\frac{x}{2} and tan⁡x2\tan\frac{x}{2}, given sin⁡x=14\sin x = \frac{1}{4}, xx in quadrant II.