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Mathematics · Textbook solutions

Trigonometric Functions

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 190 questions

3.1 Trigonometric Equations and their Solutions

50 q

Solved Examples

Worked · 23
  1. 3.1 SolvedEx.P1
    Find the principal solutions of sin⁡θ=12\sin\theta = \dfrac{1}{\sqrt{2}}.
  2. 3.1 SolvedEx.P2
    Find the principal solutions of cos⁡θ=12\cos\theta = \dfrac{1}{2}.
  3. 3.1 SolvedEx.P3
    Find the principal solutions of cos⁡θ=−12\cos\theta = -\dfrac{1}{2}.
  4. 3.1 SolvedEx.P4
    Find the principal solutions of cot⁡θ=−3\cot\theta = -\sqrt{3}.
  5. Find the general solution of the following equations.
    3.1 SolvedEx.G1 i)
    sin⁡θ=32\sin\theta = \dfrac{\sqrt{3}}{2}
  6. 3.1 SolvedEx.G1 ii)
    cos⁡θ=12\cos\theta = \dfrac{1}{\sqrt{2}}
  7. 3.1 SolvedEx.G1 iii)
    tan⁡θ=3\tan\theta = \sqrt{3}
  8. Find the general solution of the following equations.
    3.1 SolvedEx.G2 i)
    sin⁡θ=−32\sin\theta = -\dfrac{\sqrt{3}}{2}
  9. 3.1 SolvedEx.G2 ii)
    cos⁡θ=−12\cos\theta = -\dfrac{1}{2}
  10. 3.1 SolvedEx.G2 iii)
    cot⁡θ=−3\cot\theta = -\sqrt{3}
  11. Find the general solution of the following equations.
    3.1 SolvedEx.G3 i)
    cosec⁡θ=2\operatorname{cosec}\theta = 2
  12. 3.1 SolvedEx.G3 ii)
    sec⁡θ+2=0\sec\theta + \sqrt{2} = 0
  13. Find the general solution of the following equations.
    3.1 SolvedEx.G4 i)
    cos⁡2θ=−12\cos 2\theta = -\dfrac{1}{\sqrt{2}}
  14. 3.1 SolvedEx.G4 ii)
    tan⁡3θ=−1\tan 3\theta = -1
  15. 3.1 SolvedEx.G4 iii)
    sin⁡4θ=32\sin 4\theta = \dfrac{\sqrt{3}}{2}
  16. Find the general solution of the following equations.
    3.1 SolvedEx.G5 i)
    4cos⁡2θ=14\cos^2\theta = 1
  17. 3.1 SolvedEx.G5 ii)
    4sin⁡2θ=34\sin^2\theta = 3
  18. 3.1 SolvedEx.G5 iii)
    tan⁡2θ=1\tan^2\theta = 1
  19. 3.1 SolvedEx.G6
    Find the general solution of cos⁡3θ=cos⁡2θ\cos 3\theta = \cos 2\theta.
  20. 3.1 SolvedEx.G7
    Find the general solution of cos⁡5θ=sin⁡3θ\cos 5\theta = \sin 3\theta.
  21. 3.1 SolvedEx.G8
    Find the general solution of sec⁡22θ=1−tan⁡2θ\sec^2 2\theta = 1 - \tan 2\theta.
  22. 3.1 SolvedEx.G9
    Find the general solution of sin⁡θ+sin⁡3θ+sin⁡5θ=0\sin\theta + \sin 3\theta + \sin 5\theta = 0.
  23. 3.1 SolvedEx.G10
    Find the general solution of cos⁡θ−sin⁡θ=1\cos\theta - \sin\theta = 1.

Exercise 3.1

Practice · 27
  1. Find the principal solutions of the following equations.
    3.1 Ex.Q1 i)
    cos⁡θ=12\cos\theta = \dfrac{1}{2}
  2. 3.1 Ex.Q1 ii)
    sec⁡θ=23\sec\theta = \dfrac{2}{\sqrt{3}}
  3. 3.1 Ex.Q1 iii)
    cot⁡θ=3\cot\theta = \sqrt{3}
  4. 3.1 Ex.Q1 iv)
    cot⁡θ=0\cot\theta = 0
  5. Find the principal solutions of the following equations.
    3.1 Ex.Q2 i)
    sin⁡θ=−12\sin\theta = -\dfrac{1}{2}
  6. 3.1 Ex.Q2 ii)
    tan⁡θ=−1\tan\theta = -1
  7. 3.1 Ex.Q2 iii)
    3cosec⁡θ+2=0\sqrt{3}\operatorname{cosec}\theta + 2 = 0
  8. Find the general solutions of the following equations.
    3.1 Ex.Q3 i)
    sin⁡θ=12\sin\theta = \dfrac{1}{2}
  9. 3.1 Ex.Q3 ii)
    cos⁡θ=32\cos\theta = \dfrac{\sqrt{3}}{2}
  10. 3.1 Ex.Q3 iii)
    tan⁡θ=13\tan\theta = \dfrac{1}{\sqrt{3}}
  11. 3.1 Ex.Q3 iv)
    cot⁡θ=0\cot\theta = 0
  12. Find the general solutions of the following equations.
    3.1 Ex.Q4 i)
    sec⁡θ=2\sec\theta = \sqrt{2}
  13. 3.1 Ex.Q4 ii)
    cosec⁡θ=−2\operatorname{cosec}\theta = -\sqrt{2}
  14. 3.1 Ex.Q4 iii)
    tan⁡θ=−1\tan\theta = -1
  15. Find the general solutions of the following equations.
    3.1 Ex.Q5 i)
    sin⁡2θ=12\sin 2\theta = \dfrac{1}{2}
  16. 3.1 Ex.Q5 ii)
    tan⁡2θ3=3\tan\dfrac{2\theta}{3} = \sqrt{3}
  17. 3.1 Ex.Q5 iii)
    cot⁡4θ=−1\cot 4\theta = -1
  18. Find the general solutions of the following equations.
    3.1 Ex.Q6 i)
    4cos⁡2θ=34\cos^2\theta = 3
  19. 3.1 Ex.Q6 ii)
    4sin⁡2θ=14\sin^2\theta = 1
  20. 3.1 Ex.Q6 iii)
    cos⁡4θ=cos⁡2θ\cos 4\theta = \cos 2\theta
  21. Find the general solutions of the following equations.
    3.1 Ex.Q7 i)
    sin⁡θ=tan⁡θ\sin\theta = \tan\theta
  22. 3.1 Ex.Q7 ii)
    tan⁡3θ=3tan⁡θ\tan^3\theta = 3\tan\theta
  23. 3.1 Ex.Q7 iii)
    cos⁡θ+sin⁡θ=1\cos\theta + \sin\theta = 1
  24. Which of the following equations have solutions?
    3.1 Ex.Q8 i)
    cos⁡2θ=−1\cos 2\theta = -1
  25. 3.1 Ex.Q8 ii)
    cos⁡2θ=−1\cos^2\theta = -1
  26. 3.1 Ex.Q8 iii)
    2sin⁡θ=32\sin\theta = 3
  27. 3.1 Ex.Q8 iv)
    3tan⁡θ=53\tan\theta = 5

3.2 Solution of Triangle

32 q

Solved Examples

Worked · 14
  1. 3.2 SolvedEx.Polar.1
    Find the Cartesian co-ordinates of the point whose polar co-ordinates are (2,π4)\left(2, \dfrac{\pi}{4}\right).
  2. 3.2 SolvedEx.Polar.2
    Find the polar co-ordinates of the point whose Cartesian co-ordinates are (12,−12)\left(\dfrac{1}{\sqrt{2}}, \dfrac{-1}{\sqrt{2}}\right).
  3. 3.2 SolvedEx.Sine.1
    In △ABC\triangle ABC, if A=30∘A = 30^\circ, B=60∘B = 60^\circ then find the ratio of sides.
  4. 3.2 SolvedEx.Sine.2
    In △ABC\triangle ABC, if a=2a = 2, b=3b = 3 and sin⁡A=23\sin A = \dfrac{2}{3} then find BB.
  5. 3.2 SolvedEx.Sine.3
    In △ABC\triangle ABC, prove that a(sin⁡B−sin⁡C)+b(sin⁡C−sin⁡A)+c(sin⁡A−sin⁡B)=0a(\sin B - \sin C) + b(\sin C - \sin A) + c(\sin A - \sin B) = 0.
  6. 3.2 SolvedEx.Sine.4
    In △ABC\triangle ABC, prove that (a−b)sin⁡C+(b−c)sin⁡A+(c−a)sin⁡B=0(a - b)\sin C + (b - c)\sin A + (c - a)\sin B = 0.
  7. 3.2 SolvedEx.Cosine.5
    In △ABC\triangle ABC, if a=2a = 2, b=3b = 3, c=4c = 4 then prove that the triangle is obtuse angled.
  8. 3.2 SolvedEx.Cosine.6
    In △ABC\triangle ABC, if A=60∘A = 60^\circ, b=3b = 3 and c=8c = 8 then find aa. Also find the circumradius of the triangle.
  9. 3.2 SolvedEx.Cosine.7
    In △ABC\triangle ABC, prove that a(bcos⁡C−ccos⁡B)=b2−c2a(b\cos C - c\cos B) = b^2 - c^2.
  10. 3.2 SolvedEx.Projection.8
    In △ABC\triangle ABC, prove that (a+b)cos⁡C+(b+c)cos⁡A+(c+a)cos⁡B=a+b+c(a + b)\cos C + (b + c)\cos A + (c + a)\cos B = a + b + c.
  11. 3.2 SolvedEx.Projection.9
    In △ABC\triangle ABC, prove that a(cos⁡C−cos⁡B)=2(b−c)cos⁡2(A2)a(\cos C - \cos B) = 2(b - c)\cos^2\left(\dfrac{A}{2}\right).
  12. 3.2 SolvedEx.Projection.10
    Prove the Cosine rule using the Projection rule.
  13. 3.2 SolvedEx.Applications.1
    In △ABC\triangle ABC, if a=13a = 13, b=14b = 14, c=15c = 15 then find the values of (i) cos⁡A\cos A (ii) sin⁡A2\sin\dfrac{A}{2} (iii) cos⁡A2\cos\dfrac{A}{2} (iv) tan⁡A2\tan\dfrac{A}{2} (v) A(△ABC)A(\triangle ABC) (vi) sin⁡A\sin A.
  14. 3.2 SolvedEx.Applications.2
    In △ABC\triangle ABC, prove that cot⁡A2+cot⁡B2+cot⁡C2=a+b+cb+c−acot⁡A2\cot\dfrac{A}{2} + \cot\dfrac{B}{2} + \cot\dfrac{C}{2} = \dfrac{a + b + c}{b + c - a}\cot\dfrac{A}{2}.

Exercise 3.2

Practice · 18
  1. Find the Cartesian co-ordinates of the point whose polar co-ordinates are :
    3.2 Q1 i)
    (2,π4)\left(\sqrt{2}, \dfrac{\pi}{4}\right)
  2. 3.2 Q1 ii)
    (4,π2)\left(4, \dfrac{\pi}{2}\right)
  3. 3.2 Q1 iii)
    (34,3π4)\left(\dfrac{3}{4}, \dfrac{3\pi}{4}\right)
  4. 3.2 Q1 iv)
    (12,7π3)\left(\dfrac{1}{2}, \dfrac{7\pi}{3}\right)
  5. Find the polar co-ordinates of the point whose Cartesian co-ordinates are :
    3.2 Q2 i)
    (2,2)\left(\sqrt{2}, \sqrt{2}\right)
  6. 3.2 Q2 ii)
    (0,12)\left(0, \dfrac{1}{2}\right)
  7. 3.2 Q2 iii)
    (1,−3)\left(1, -\sqrt{3}\right)
  8. 3.2 Q2 iv)
    (32,332)\left(\dfrac{3}{2}, \dfrac{3\sqrt{3}}{2}\right)
  9. 3.2 Q3
    In △ABC\triangle ABC, if A=45∘A = 45^\circ, B=60∘B = 60^\circ then find the ratio of its sides.
  10. 3.2 Q4
    In △ABC\triangle ABC, prove that sin⁡(B−C2)=(b−ca)cos⁡A2\sin\left(\dfrac{B - C}{2}\right) = \left(\dfrac{b - c}{a}\right)\cos\dfrac{A}{2}.
  11. 3.2 Q5
    With usual notations prove that 2{asin⁡2C2+csin⁡2A2}=a−b+c2\left\{a\sin^2\dfrac{C}{2} + c\sin^2\dfrac{A}{2}\right\} = a - b + c.
  12. 3.2 Q6
    In △ABC\triangle ABC, prove that a3sin⁡(B−C)+b3sin⁡(C−A)+c3sin⁡(A−B)=0a^3\sin(B - C) + b^3\sin(C - A) + c^3\sin(A - B) = 0.
  13. 3.2 Q7
    In △ABC\triangle ABC, if cot⁡A\cot A, cot⁡B\cot B, cot⁡C\cot C are in A.P. then show that a2a^2, b2b^2, c2c^2 are also in A.P.
  14. 3.2 Q8
    In △ABC\triangle ABC, if acos⁡A=bcos⁡Ba\cos A = b\cos B then prove that the triangle is right angled or an isosceles triangle.
  15. 3.2 Q9
    With usual notations prove that 2(bccos⁡A+accos⁡B+abcos⁡C)=a2+b2+c22(bc\cos A + ac\cos B + ab\cos C) = a^2 + b^2 + c^2.
  16. 3.2 Q10
    In △ABC\triangle ABC, if a=18a = 18, b=24b = 24, c=30c = 30 then find the values of (i) cos⁡A\cos A (ii) sin⁡A2\sin\dfrac{A}{2} (iii) cos⁡A2\cos\dfrac{A}{2} (iv) tan⁡A2\tan\dfrac{A}{2} (v) A(△ABC)A(\triangle ABC) (vi) sin⁡A\sin A.
  17. 3.2 Q11
    In △ABC\triangle ABC, prove that (b+c−a)tan⁡A2=(c+a−b)tan⁡B2=(a+b−c)tan⁡C2(b + c - a)\tan\dfrac{A}{2} = (c + a - b)\tan\dfrac{B}{2} = (a + b - c)\tan\dfrac{C}{2}.
  18. 3.2 Q12
    In △ABC\triangle ABC, prove that sin⁡A2sin⁡B2sin⁡C2=[A(△ABC)]2abcs\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2} = \dfrac{[A(\triangle ABC)]^2}{abcs}.

3.3 Inverse Trigonometric Functions

32 q

Solved Examples

Worked · 14
  1. Find the principal values of the following :
    3.3 SolvedEx.1 i)
    sin⁡−1(−12)\sin^{-1}\left(-\frac{1}{2}\right)
  2. 3.3 SolvedEx.1 ii)
    cos⁡−1(32)\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)
  3. 3.3 SolvedEx.1 iii)
    cot⁡−1(−13)\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)
  4. Find the values of the following :
    3.3 SolvedEx.2 i)
    sin⁡−1(sin⁡5π3)\sin^{-1}\left(\sin\frac{5\pi}{3}\right)
  5. 3.3 SolvedEx.2 ii)
    tan⁡−1(tan⁡π4)\tan^{-1}\left(\tan\frac{\pi}{4}\right)
  6. 3.3 SolvedEx.2 iii)
    sin⁡(cos⁡−1(−12))\sin\left(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\right)
  7. 3.3 SolvedEx.2 iv)
    sin⁡(cos⁡−145+tan⁡−1512)\sin\left(\cos^{-1}\frac{4}{5} + \tan^{-1}\frac{5}{12}\right)
  8. Find the values of the following :
    3.3 SolvedEx.3 i)
    sin⁡[sin⁡−1(35)+cos⁡−1(35)]\sin\left[\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{3}{5}\right)\right]
  9. 3.3 SolvedEx.3 ii)
    cos⁡[cos⁡−1(−12)+tan⁡−13]\cos\left[\cos^{-1}\left(-\frac{1}{2}\right) + \tan^{-1}\sqrt{3}\right]
  10. 3.3 SolvedEx.4
    If ∣x∣≤1|x| \le 1, show that sin⁡(cos⁡−1x)=cos⁡(sin⁡−1x)\sin(\cos^{-1}x) = \cos(\sin^{-1}x).
  11. Prove the following :
    3.3 SolvedEx.5 i)
    2tan⁡−1(−13)+cos⁡−1(35)=π22\tan^{-1}\left(-\frac{1}{3}\right) + \cos^{-1}\left(\frac{3}{5}\right) = \frac{\pi}{2}
  12. 3.3 SolvedEx.5 ii)
    2tan⁡−1(13)+tan⁡−1(17)=π42\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) = \frac{\pi}{4}
  13. 3.3 SolvedEx.6
    Prove that tan⁡−11+tan⁡−12+tan⁡−13=π\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi.
  14. 3.3 SolvedEx.7
    Prove that cos⁡−145+cos⁡−11213=cos⁡−13365\cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} = \cos^{-1}\frac{33}{65}.

Exercise 3.3

Practice · 18
  1. Find the principal values of the following :
    Q.1 i)
    sin⁡−1(12)\sin^{-1}\left(\frac{1}{2}\right)
  2. Q.1 ii)
    csc⁡−1(2)\csc^{-1}(2)
  3. Q.1 iii)
    tan⁡−1(−1)\tan^{-1}(-1)
  4. Q.1 iv)
    tan⁡−1(−3)\tan^{-1}\left(-\sqrt{3}\right)
  5. Q.1 v)
    sin⁡−1(12)\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)
  6. Q.1 vi)
    cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right)
  7. Evaluate the following :
    Q.2 i)
    tan⁡−1(1)+cos⁡−1(12)+sin⁡−1(12)\tan^{-1}(1) + \cos^{-1}\left(\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)
  8. Q.2 ii)
    cos⁡−1(12)+2sin⁡−1(12)\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right)
  9. Q.2 iii)
    tan⁡−13−sec⁡−1(−2)\tan^{-1}\sqrt{3} - \sec^{-1}(-2)
  10. Q.2 iv)
    csc⁡−1(−2)+cot⁡−1(3)\csc^{-1}\left(-\sqrt{2}\right) + \cot^{-1}\left(\sqrt{3}\right)
  11. Prove the following :
    Q.3 i)
    sin⁡−1(12)−3sin⁡−1(32)=−3π4\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = -\frac{3\pi}{4}
  12. Q.3 ii)
    sin⁡−1(−12)+cos⁡−1(−32)=cos⁡−1(−12)\sin^{-1}\left(-\frac{1}{2}\right) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \cos^{-1}\left(-\frac{1}{2}\right)
  13. Q.3 iii)
    sin⁡−1(35)+cos⁡−1(1213)=sin⁡−1(5665)\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{56}{65}\right)
  14. Q.3 iv)
    cos⁡−1(35)+cos⁡−1(45)=π2\cos^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{4}{5}\right) = \frac{\pi}{2}
  15. Q.3 v)
    tan⁡−1(12)+tan⁡−1(13)=π4\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4}
  16. Q.3 vi)
    2tan⁡−1(13)=tan⁡−1(34)2\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{3}{4}\right)
  17. Q.3 vii)
    tan⁡−1[cos⁡θ+sin⁡θcos⁡θ−sin⁡θ]=π4+θ\tan^{-1}\left[\frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta}\right] = \frac{\pi}{4} + \theta if θ∈(−π4,π4)\theta \in \left(-\frac{\pi}{4},\frac{\pi}{4}\right)
  18. Q.3 viii)
    tan⁡−11−cos⁡θ1+cos⁡θ=θ2\tan^{-1}\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \frac{\theta}{2}, if θ∈(0,π)\theta \in (0,\pi)

Miscellaneous Exercise 3

76 q

Select the correct option

Practice · 20
  1. Misc I Q.1
    The principal solutions of the equation sin⁡θ=−12\sin\theta = \dfrac{-1}{2} are ______.

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  2. Misc I Q.2
    The principal solutions of the equation cot⁡θ=3\cot\theta = \sqrt{3} are ______.

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  3. Misc I Q.3
    The general solution of sec⁡x=2\sec x = \sqrt{2} is ______.

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  4. Misc I Q.4
    If cos⁡pθ=cos⁡qθ, p≠q\cos p\theta = \cos q\theta,\ p \neq q then ______.

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  5. Misc I Q.5
    If the polar co-ordinates of a point are (2,π4)\left(2, \dfrac{\pi}{4}\right) then its cartesian co-ordinates are ______.

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  6. Misc I Q.6
    If 3cos⁡x−sin⁡x=1\sqrt{3}\cos x - \sin x = 1, then the general value of xx is ______.

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  7. Misc I Q.7
    In △ABC\triangle ABC, if ∠A=45∘, ∠B=30∘\angle A = 45^{\circ},\ \angle B = 30^{\circ}, then the ratio a:b:c=a : b : c = ______.

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  8. Misc I Q.8
    In △ABC\triangle ABC, if c2+a2−b2=acc^{2} + a^{2} - b^{2} = ac, then ∠B=\angle B = ______.

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  9. Misc I Q.9
    In △ABC\triangle ABC, accos⁡B−bccos⁡A=ac\cos B - bc\cos A = ______.

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  10. Misc I Q.10
    If in a triangle the angles are in A.P. and b:c=3:2b : c = \sqrt{3} : \sqrt{2}, then AA is equal to ______.

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  11. Misc I Q.11
    cos⁡−1(cos⁡7π6)=\cos^{-1}\left(\cos\dfrac{7\pi}{6}\right) = ______.

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  12. Misc I Q.12
    The principal value of sin⁡−1(−32)\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right) is ______.

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  13. Misc I Q.13
    If sin⁡−145+cos⁡−11213=sin⁡−1α\sin^{-1}\dfrac{4}{5} + \cos^{-1}\dfrac{12}{13} = \sin^{-1}\alpha, then α=\alpha = ______.

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  14. Misc I Q.14
    If tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}, then x=x = ______.

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  15. Misc I Q.15
    2tan⁡−1(13)+tan⁡−1(17)=2\tan^{-1}\left(\dfrac{1}{3}\right) + \tan^{-1}\left(\dfrac{1}{7}\right) = ______.

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  16. Misc I Q.16
    tan⁡[2tan⁡−1(15)−π4]=\tan\left[2\tan^{-1}\left(\dfrac{1}{5}\right) - \dfrac{\pi}{4}\right] = ______.

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  17. Misc I Q.17
    The principal value branch of sec⁡−1x\sec^{-1}x is ______.

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  18. Misc I Q.18
    cos⁡[tan⁡−113+tan⁡−112]=\cos\left[\tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{2}\right] = ______.

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  19. Misc I Q.19
    If tan⁡θ+tan⁡2θ+tan⁡3θ=tan⁡θ tan⁡2θ tan⁡3θ\tan\theta + \tan 2\theta + \tan 3\theta = \tan\theta\,\tan 2\theta\,\tan 3\theta, then the general value of θ\theta is ______.

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  20. Misc I Q.20
    In any △ABC\triangle ABC, if acos⁡B=bcos⁡Aa\cos B = b\cos A, then the triangle is ______.

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Solve the following

Practice · 56
  1. Find the principal solutions of the following equations:
    Misc II Q.1 (i)
    sin⁡2θ=−12\sin 2\theta = -\dfrac{1}{2}
  2. Find the principal solutions of the following equations:
    Misc II Q.1 (ii)
    tan⁡3θ=−1\tan 3\theta = -1
  3. Find the principal solutions of the following equations:
    Misc II Q.1 (iii)
    cot⁡θ=0\cot\theta = 0
  4. Find the principal solutions of the following equations:
    Misc II Q.2 (i)
    sin⁡2θ=−12\sin 2\theta = -\dfrac{1}{\sqrt{2}}
  5. Find the principal solutions of the following equations:
    Misc II Q.2 (ii)
    tan⁡5θ=−1\tan 5\theta = -1
  6. Find the principal solutions of the following equations:
    Misc II Q.2 (iii)
    cot⁡2θ=0\cot 2\theta = 0
  7. Which of the following equations have no solutions?
    Misc II Q.3 (i)
    cos⁡2θ=13\cos 2\theta = \dfrac{1}{3}
  8. Which of the following equations have no solutions?
    Misc II Q.3 (ii)
    cos⁡2θ=−1\cos^{2}\theta = -1
  9. Which of the following equations have no solutions?
    Misc II Q.3 (iii)
    2sin⁡θ=32\sin\theta = 3
  10. Which of the following equations have no solutions?
    Misc II Q.3 (iv)
    3sin⁡θ=53\sin\theta = 5
  11. Find the general solutions of the following equations:
    Misc II Q.4 (i)
    tan⁡θ=−3\tan\theta = -\sqrt{3}
  12. Find the general solutions of the following equations:
    Misc II Q.4 (ii)
    tan⁡2θ=3\tan^{2}\theta = 3
  13. Find the general solutions of the following equations:
    Misc II Q.4 (iii)
    sin⁡θ−cos⁡θ=1\sin\theta - \cos\theta = 1
  14. Find the general solutions of the following equations:
    Misc II Q.4 (iv)
    sin⁡2θ−cos⁡2θ=1\sin^{2}\theta - \cos^{2}\theta = 1
  15. Misc II Q.5
    In △ABC\triangle ABC prove that cos⁡(A−B2)=(a+bc)sin⁡C2\cos\left(\dfrac{A-B}{2}\right) = \left(\dfrac{a+b}{c}\right)\sin\dfrac{C}{2}.
  16. Misc II Q.6
    With usual notations prove that sin⁡(A−B)sin⁡(A+B)=a2−b2c2\dfrac{\sin(A-B)}{\sin(A+B)} = \dfrac{a^{2}-b^{2}}{c^{2}}.
  17. Misc II Q.7
    In △ABC\triangle ABC prove that (a−b)2cos⁡2C2+(a+b)2sin⁡2C2=c2(a-b)^{2}\cos^{2}\dfrac{C}{2} + (a+b)^{2}\sin^{2}\dfrac{C}{2} = c^{2}.
  18. Misc II Q.8
    In △ABC\triangle ABC, if cos⁡A=sin⁡B−cos⁡C\cos A = \sin B - \cos C, then show that it is a right angled triangle.
  19. Misc II Q.9
    If sin⁡Asin⁡C=sin⁡(A−B)sin⁡(B−C)\dfrac{\sin A}{\sin C} = \dfrac{\sin(A-B)}{\sin(B-C)}, then show that a2,b2,c2a^{2}, b^{2}, c^{2} are in A.P.
  20. Misc II Q.10
    Solve the triangle in which a=3+1, b=3−1a = \sqrt{3}+1,\ b = \sqrt{3}-1 and C=60∘C = 60^{\circ}.
  21. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (i)
    asin⁡A−bsin⁡B=csin⁡(A−B)a\sin A - b\sin B = c\sin(A-B)
  22. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (ii)
    c−bcos⁡Ab−ccos⁡A=cos⁡Bcos⁡C\dfrac{c - b\cos A}{b - c\cos A} = \dfrac{\cos B}{\cos C}
  23. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (iii)
    a2sin⁡(B−C)=(b2−c2)sin⁡Aa^{2}\sin(B-C) = (b^{2}-c^{2})\sin A
  24. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (iv)
    accos⁡B−bccos⁡A=(a2−b2)ac\cos B - bc\cos A = (a^{2}-b^{2})
  25. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (v)
    cos⁡Aa+cos⁡Bb+cos⁡Cc=a2+b2+c22abc\dfrac{\cos A}{a} + \dfrac{\cos B}{b} + \dfrac{\cos C}{c} = \dfrac{a^{2}+b^{2}+c^{2}}{2abc}
  26. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (vi)
    cos⁡2Aa2−cos⁡2Bb2=1a2−1b2\dfrac{\cos 2A}{a^{2}} - \dfrac{\cos 2B}{b^{2}} = \dfrac{1}{a^{2}} - \dfrac{1}{b^{2}}
  27. In △ABC\triangle ABC prove the following:
    Misc II Q.11 (vii)
    b−ca=tan⁡B2−tan⁡C2tan⁡B2+tan⁡C2\dfrac{b-c}{a} = \dfrac{\tan\dfrac{B}{2} - \tan\dfrac{C}{2}}{\tan\dfrac{B}{2} + \tan\dfrac{C}{2}}
  28. Misc II Q.12
    In △ABC\triangle ABC, if a2,b2,c2a^{2}, b^{2}, c^{2} are in A.P., then cot⁡A2,cot⁡B2,cot⁡C2\cot\dfrac{A}{2}, \cot\dfrac{B}{2}, \cot\dfrac{C}{2} are also in A.P.
  29. Misc II Q.13
    In △ABC\triangle ABC, if C=90∘C = 90^{\circ}, then prove that sin⁡(A−B)=a2−b2a2+b2\sin(A-B) = \dfrac{a^{2}-b^{2}}{a^{2}+b^{2}}.
  30. Misc II Q.14
    In △ABC\triangle ABC, if cos⁡Aa=cos⁡Bb\dfrac{\cos A}{a} = \dfrac{\cos B}{b}, then show that it is an isosceles triangle.
  31. Misc II Q.15
    In △ABC\triangle ABC, if sin⁡2A+sin⁡2B=sin⁡2C\sin^{2}A + \sin^{2}B = \sin^{2}C, then prove that the triangle is a right angled triangle.
  32. Misc II Q.16
    In △ABC\triangle ABC prove that a2(cos⁡2B−cos⁡2C)+b2(cos⁡2C−cos⁡2A)+c2(cos⁡2A−cos⁡2B)=0a^{2}(\cos^{2}B - \cos^{2}C) + b^{2}(\cos^{2}C - \cos^{2}A) + c^{2}(\cos^{2}A - \cos^{2}B) = 0.
  33. Misc II Q.17
    With usual notations show that (c2−a2+b2)tan⁡A=(a2−b2+c2)tan⁡B=(b2−c2+a2)tan⁡C(c^{2} - a^{2} + b^{2})\tan A = (a^{2} - b^{2} + c^{2})\tan B = (b^{2} - c^{2} + a^{2})\tan C.
  34. Misc II Q.18
    In △ABC\triangle ABC, if acos⁡2C2+ccos⁡2A2=3b2a\cos^{2}\dfrac{C}{2} + c\cos^{2}\dfrac{A}{2} = \dfrac{3b}{2}, then prove that a,b,ca, b, c are in A.P.
  35. Misc II Q.19
    Show that 2sin⁡−135=tan⁡−12472\sin^{-1}\dfrac{3}{5} = \tan^{-1}\dfrac{24}{7}.
  36. Misc II Q.20
    Show that tan⁡−115+tan⁡−117+tan⁡−113+tan⁡−118=π4\tan^{-1}\dfrac{1}{5} + \tan^{-1}\dfrac{1}{7} + \tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{8} = \dfrac{\pi}{4}.
  37. Misc II Q.21
    Prove that tan⁡−1x=12cos⁡−1(1−x1+x)\tan^{-1}\sqrt{x} = \dfrac{1}{2}\cos^{-1}\left(\dfrac{1-x}{1+x}\right) if x∈[0,1]x \in [0,1].
  38. Misc II Q.22
    Show that 9π5−94sin⁡−113=94sin⁡−1223\dfrac{9\pi}{5} - \dfrac{9}{4}\sin^{-1}\dfrac{1}{3} = \dfrac{9}{4}\sin^{-1}\dfrac{2\sqrt{2}}{3}.
  39. Misc II Q.23
    Show that tan⁡−1(1+x−1−x1+x+1−x)=π4−12cos⁡−1x, for −12≤x≤1\tan^{-1}\left(\dfrac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \dfrac{\pi}{4} - \dfrac{1}{2}\cos^{-1}x,\ \text{for } -\dfrac{1}{\sqrt{2}} \le x \le 1.
  40. Misc II Q.24
    If sin⁡(sin⁡−115+cos⁡−1x)=1\sin\left(\sin^{-1}\dfrac{1}{5} + \cos^{-1}x\right) = 1, then find the value of xx.
  41. Misc II Q.25
    If tan⁡−1(x−1x−2)+tan⁡−1(x+1x+2)=π4\tan^{-1}\left(\dfrac{x-1}{x-2}\right) + \tan^{-1}\left(\dfrac{x+1}{x+2}\right) = \dfrac{\pi}{4}, then find the value of xx.
  42. Misc II Q.26
    If 2tan⁡−1(cos⁡x)=tan⁡−1(csc⁡x)2\tan^{-1}(\cos x) = \tan^{-1}(\csc x), then find the value of xx.
  43. Misc II Q.27
    Solve: tan⁡−1(1−x1+x)=12tan⁡−1x, for x>0\tan^{-1}\left(\dfrac{1-x}{1+x}\right) = \dfrac{1}{2}\tan^{-1}x,\ \text{for } x > 0.
  44. Misc II Q.28
    If sin⁡−1(1−x)−2sin⁡−1x=π2\sin^{-1}(1-x) - 2\sin^{-1}x = \dfrac{\pi}{2}, then find the value of xx.
  45. Misc II Q.29
    If tan⁡−12x+tan⁡−13x=π2\tan^{-1}2x + \tan^{-1}3x = \dfrac{\pi}{2}, then find the value of xx.
  46. Misc II Q.30
    Show that tan⁡−112−tan⁡−114=tan⁡−129\tan^{-1}\dfrac{1}{2} - \tan^{-1}\dfrac{1}{4} = \tan^{-1}\dfrac{2}{9}.
  47. Misc II Q.31
    Show that cot⁡−113−tan⁡−113=cot⁡−134\cot^{-1}\dfrac{1}{3} - \tan^{-1}\dfrac{1}{3} = \cot^{-1}\dfrac{3}{4}.
  48. Misc II Q.32
    Show that tan⁡−112=13tan⁡−1112\tan^{-1}\dfrac{1}{2} = \dfrac{1}{3}\tan^{-1}\dfrac{11}{2}.
  49. Misc II Q.33
    Show that cos⁡−132+2sin⁡−132=5π6\cos^{-1}\dfrac{\sqrt{3}}{2} + 2\sin^{-1}\dfrac{\sqrt{3}}{2} = \dfrac{5\pi}{6}.
  50. Misc II Q.34
    Show that 2cot⁡−132+sec⁡−11312=π22\cot^{-1}\dfrac{3}{2} + \sec^{-1}\dfrac{13}{12} = \dfrac{\pi}{2}.
  51. Prove the following:
    Misc II Q.35 (i)
    cos⁡−1x=tan⁡−11−x2x if x<0\cos^{-1}x = \tan^{-1}\dfrac{\sqrt{1-x^{2}}}{x}\ \text{if } x < 0.
  52. Prove the following:
    Misc II Q.35 (ii)
    cos⁡−1x=π+tan⁡−11−x2x if x<0\cos^{-1}x = \pi + \tan^{-1}\dfrac{\sqrt{1-x^{2}}}{x}\ \text{if } x < 0.
  53. Misc II Q.36
    If ∣x∣<1|x| < 1, then prove that 2tan⁡−1x=tan⁡−12x1−x2=sin⁡−12x1+x2=cos⁡−11−x21+x22\tan^{-1}x = \tan^{-1}\dfrac{2x}{1-x^{2}} = \sin^{-1}\dfrac{2x}{1+x^{2}} = \cos^{-1}\dfrac{1-x^{2}}{1+x^{2}}.
  54. Misc II Q.37
    If x,y,zx, y, z are positive then prove that tan⁡−1x−y1+xy+tan⁡−1y−z1+yz+tan⁡−1z−x1+zx=0\tan^{-1}\dfrac{x-y}{1+xy} + \tan^{-1}\dfrac{y-z}{1+yz} + \tan^{-1}\dfrac{z-x}{1+zx} = 0.
  55. Misc II Q.38
    If tan⁡−1x+tan⁡−1y+tan⁡−1z=π2\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \dfrac{\pi}{2}, then show that xy+yz+zx=1xy + yz + zx = 1.
  56. Misc II Q.39
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=π\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi, then show that x2+y2+z2+2xyz=1x^{2} + y^{2} + z^{2} + 2xyz = 1.