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Mathematics · Class 12 Science

Ch 3Trigonometric Functions — Class 12 Mathematics, concept-first.

We are already familiar with algebraic equations such as , which are satisfied by only finitely many values of . In this chapter we turn instead to trigonometric equations -- equations involving one or more trigonometric functions of an unknown angle.

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Trigonometric Equations and Principal Solutions

A trigonometric equation is any equation involving one or more trigonometric functions of an unknown angle, such as or .

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Introduction

We are already familiar with algebraic equations such as , which are satisfied by only finitely many values of .

3.1

Trigonometric Equations and their Solutions

An algebraic equation like is satisfied by only finitely many values of . A trigonometric equation is an equation that involves one or more trigonometric functions of an unknown angle, for example , ,…

+Exercise 3.1i24 questions
  1. Q1Find the principal solutions of the following equation: $\cos\theta = \frac{1}{2}$Free
  2. Q2Find the principal solutions of the following equation: $\sec\theta = \frac{2}{\sqrt{3}}$Free
  3. Q3Find the principal solutions of the following equation: $\cot\theta = \sqrt{3}$Free
  4. Q4Find the principal solutions of the following equation: $\cot\theta = 0$Preview
  5. Q5Find the principal solutions of the following equation: $\sin\theta = -\frac{1}{2}$Preview
  6. Q6Find the principal solutions of the following equation: $\tan\theta = -1$Preview
  7. Q7Find the principal solutions of the following equation: $\sqrt{3}\,\text{cosec}\,\theta + 2 = 0$Preview
  8. Q8Find the general solution of the following equation: $\sin\theta = \frac{1}{2}$Preview
  9. Q9Find the general solution of the following equation: $\cos\theta = \frac{\sqrt{3}}{2}$Preview
  10. Q10Find the general solution of the following equation: $\tan\theta = \frac{1}{\sqrt{3}}$Preview
  11. Q11Find the general solution of the following equation: $\cot\theta = 0$Preview
  12. Q12Find the general solution of the following equation: $\sec\theta = \sqrt{2}$Preview
  13. Q13Find the general solution of the following equation: $\text{cosec}\,\theta = -2$Preview
  14. Q14Find the general solution of the following equation: $\tan\theta = -1$Preview
  15. Q15Find the general solution of the following equation: $\sin 2\theta = \frac{1}{2}$Preview
  16. Q16Find the general solution of the following equation: $\tan\dfrac{2\theta}{3} = \sqrt{3}$Preview
  17. Q17Find the general solution of the following equation: $\cot 4\theta = -1$Preview
  18. Q18Find the general solution of the following equation: $4\cos^2\theta = 3$Preview
  19. Q19Find the general solution of the following equation: $4\sin^2\theta = 1$Preview
  20. Q20Find the general solution of the following equation: $\cos 4\theta = \cos 2\theta$Preview
  21. Q21Find the general solution of the following equation: $\sin\theta = \tan\theta$Preview
  22. Q22Find the general solution of the following equation: $\tan 3\theta = 3\tan\theta$Preview
  23. Q23Find the general solution of the following equation: $\cos\theta + \sin\theta = 1$Preview
  24. Q24Which of the following equations have solutions? (i) $\cos 2\theta = -1$ (ii) $\cos^2\theta = -1$ (iii) $2\sin\theta = 3$ (iv) $3\tan\theta…Preview
3.1.1

Trigonometric Equations and Principal Solutions

A solution of a trigonometric equation is any value of the angle that makes the equation true when substituted in. For example, satisfies because ; so does , because as well.

3.1.2

The General Solution

Since a trigonometric equation has infinitely many solutions repeating every period, we want one formula — parametrised by an integer — that generates every solution. This is the general solution.

3.2

Solution of Triangle

Having solved trigonometric equations, we now turn to a second classical application of trigonometry: relating the angles and sides of a triangle to each other so precisely that knowing any three elem…

+Exercise 3.2i23 questions
  1. Q25Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(2, \dfrac{\pi}{4}\right)$Free
  2. Q26Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(4, \dfrac{\pi}{2}\right)$Free
  3. Q27Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(\dfrac{3}{4}, \dfrac{3\pi}{4}\right)$Free
  4. Q28Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(\dfrac{1}{2}, \dfrac{7\pi}{3}\right)$Preview
  5. Q29Find the polar co-ordinates of the point whose Cartesian co-ordinates are $\left(\sqrt{2}, \sqrt{2}\right)$Preview
  6. Q30Find the polar co-ordinates of the point whose Cartesian co-ordinates are $\left(0, \dfrac{1}{2}\right)$Preview
  7. Q31Find the polar co-ordinates of the point whose Cartesian co-ordinates are $\left(1, -\sqrt{3}\right)$Preview
  8. Q32Find the polar co-ordinates of the point whose Cartesian co-ordinates are $\left(\dfrac{3}{2}, \dfrac{3\sqrt{3}}{2}\right)$Preview
  9. Q33In $\triangle ABC$, if $A = 45^\circ$, $B = 60^\circ$ then find the ratio of its sides.Preview
  10. Q34In $\triangle ABC$, prove that $\sin\left(\dfrac{B-C}{2}\right) = \dfrac{b-c}{a}\cos\dfrac{A}{2}$.Preview
  11. Q35With usual notations prove that $2\left[a\sin^2\dfrac{C}{2} + c\sin^2\dfrac{A}{2}\right] = a - b + c$.Preview
  12. Q36In $\triangle ABC$, prove that $a^3\sin(B-C) + b^3\sin(C-A) + c^3\sin(A-B) = 0$.Preview
  13. Q37In $\triangle ABC$, if $\cot A$, $\cot B$, $\cot C$ are in A.P. then show that $a^2$, $b^2$, $c^2$ are also in A.P.Preview
  14. Q38In $\triangle ABC$, if $a\cos A = b\cos B$ then prove that the triangle is right angled or an isosceles triangle.Preview
  15. Q39With usual notations prove that $2(bc\cos A + ac\cos B + ab\cos C) = a^2 + b^2 + c^2$.Preview
  16. Q40In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $\cos A$.Preview
  17. Q41In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $\sin\dfrac{A}{2}$.Preview
  18. Q42In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $\cos\dfrac{A}{2}$.Preview
  19. Q43In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $\tan\dfrac{A}{2}$.Preview
  20. Q44In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $A(\triangle ABC)$.Preview
  21. Q45In $\triangle ABC$, if $a = 18$, $b = 24$, $c = 30$ then find the value of $\sin A$.Preview
  22. Q46In $\triangle ABC$, prove that $(b+c-a)\tan\dfrac{A}{2} = (c+a-b)\tan\dfrac{B}{2} = (a+b-c)\tan\dfrac{C}{2}$.Preview
  23. Q47In $\triangle ABC$, prove that $\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2} = \dfrac{[A(\triangle ABC)]^2}{abcs}$.Preview
3.2.1

Polar Co-ordinates

Fix a point in a plane, called the pole, and a fixed ray starting at , called the polar axis. For any point in the plane other than , let (the length of the segment joining and ) and let (the angle th…

3.2.2

Relation between the Cartesian and the Polar Co-ordinates

To connect polar co-ordinates to the familiar Cartesian system, take the polar axis as the X-axis and the line through perpendicular to as the Y-axis, with the pole as the origin.

3.2.3

Solving a Triangle

The three sides and three angles of a triangle are together called its elements. Given any three of the six elements, provided at least one of them is a side, the remaining three elements can be found…

3.2.4

The Sine Rule

The Sine Rule. In , where is the circumradius (the radius of the circle passing through , , ).

3.2.5

The Cosine Rule

The Cosine Rule. In : (i) ; (ii) ; (iii) .

3.2.6

The Projection Rule

The Projection Rule. In : (i) ; (ii) ; (iii) .

3.2.7

Applications of the Sine Rule, the Cosine Rule and the Projection Rule

This section develops three important toolkits built from the Sine, Cosine, and Projection Rules. Throughout, (so is the semi-perimeter).

3.3

Inverse Trigonometric Functions

So far we have treated , , , etc. as functions that take an angle and produce a number. This section asks the reverse question: given the number, can we recover the angle? Recall that a function has a…

+Exercise 3.3i18 questions
  1. Q48Find the principal value of $\sin^{-1}\dfrac{1}{2}$Free
  2. Q49Find the principal value of $\text{cosec}^{-1}(2)$Free
  3. Q50Find the principal value of $\tan^{-1}(-1)$Free
  4. Q51Find the principal value of $\tan^{-1}(-\sqrt{3})$Preview
  5. Q52Find the principal value of $\sin^{-1}\left(-\dfrac{1}{2}\right)$Preview
  6. Q53Find the principal value of $\cos^{-1}\left(-\dfrac{1}{2}\right)$Preview
  7. Q54Evaluate: $\tan^{-1}(1) + \cos^{-1}\dfrac{1}{2} + \sin^{-1}\dfrac{1}{2}$Preview
  8. Q55Evaluate: $\cos^{-1}\dfrac{1}{2} + 2\sin^{-1}\dfrac{1}{2}$Preview
  9. Q56Evaluate: $\tan^{-1}\sqrt{3} - \sec^{-1}(-2)$Preview
  10. Q57Evaluate: $\text{cosec}^{-1}(-\sqrt{2}) + \cot^{-1}(\sqrt{3})$Preview
  11. Q58Prove the following: $\sin^{-1}\dfrac{1}{\sqrt{2}} - 3\sin^{-1}\dfrac{\sqrt{3}}{2} = -\dfrac{3\pi}{4}$Preview
  12. Q59Prove the following: $\sin^{-1}\left(-\dfrac{1}{2}\right) + \cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right) = \cos^{-1}\left(-\dfrac{1}{2}\right)…Preview
  13. Q60Prove the following: $\sin^{-1}\dfrac{3}{5} + \cos^{-1}\dfrac{12}{13} = \sin^{-1}\dfrac{56}{65}$Preview
  14. Q61Prove the following: $\cos^{-1}\dfrac{3}{5} + \cos^{-1}\dfrac{4}{5} = \dfrac{\pi}{2}$Preview
  15. Q62Prove the following: $\tan^{-1}\dfrac{1}{2} + \tan^{-1}\dfrac{1}{3} = \dfrac{\pi}{4}$Preview
  16. Q63Prove the following: $2\tan^{-1}\dfrac{1}{3} = \tan^{-1}\dfrac{3}{4}$Preview
  17. Q64Prove the following: $\tan^{-1}\left(\dfrac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta}\right) = \dfrac{\pi}{4} + \theta$, if $\theta…Preview
  18. Q65Prove the following: $\tan^{-1}\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}} = \dfrac{\theta}{2}$, if $\theta \in (0, \pi)$Preview
3.3.1

Inverse Sine Function

Consider restricted to . Graphically this restriction is one-one (strictly increasing) and onto , so its inverse exists, called the inverse sine function and denoted .

3.3.2

Inverse Cosine Function

Consider restricted to . Graphically this restriction is one-one (strictly decreasing) and onto , so its inverse exists, called the inverse cosine function, denoted .

3.3.3

Inverse Tangent Function

Consider restricted to . Graphically this restriction is one-one and onto , so its inverse exists, called the inverse tangent function, denoted . For and , we write if .

3.3.4

Inverse Cosecant Function

Consider restricted to . Graphically this restriction is one-one and onto , so its inverse exists, called the inverse cosecant function, denoted . For and , we write if .

3.3.5

Inverse Secant Function

Consider restricted to . Graphically this restriction is one-one and onto , so its inverse exists, called the inverse secant function, denoted . For and , we write if .

3.3.6

Inverse Cotangent Function

Consider restricted to . Graphically this restriction is one-one and onto , so its inverse exists, called the inverse cotangent function, denoted . For and , we write if .

3.3.7

Principal Values of Inverse Trigonometric Functions

This short section gathers the domain and principal-value range of all six inverse trigonometric functions in one place, now that each has been defined individually in the six preceding sub-sections:…

3.3.8

Properties of Inverse Trigonometric Functions

Properties of Inverse Trigonometric Functions.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 39 questions39 questions
  1. Q1If $\sin^{-1}(1-x) - 2\sin^{-1}x = \dfrac{\pi}{2}$ then $x$ is (a) $-\dfrac{1}{2}$ (b) $1$ (c) $0$ (d) $\dfrac{1}{2}$Preview
  2. Q2In $\triangle ABC$ with the usual notations prove that $(a - b)^2 \cos^2\left(\dfrac{C}{2}\right) + (a + b)^2 \sin^2\left(\dfrac{C}{2}\right…Preview
  3. Q3Find the general solution of $\sin x + \sin 3x + \sin 5x = 0$.Preview
  4. Q4In $\triangle ABC$, if $a = 13$, $b = 14$ and $c = 15$, then $\sin\left(\dfrac{A}{2}\right) =$ (a) $\dfrac{1}{5}$ (b) $\sqrt{\dfrac{1}{5}}$…Preview
  5. Q5In $\triangle ABC$, prove that $a(b\cos C - c\cos B) = b^2 - c^2$.Preview
  6. Q6Find the general solution of the equation $\sin 2x + \sin 4x + \sin 6x = 0$Preview
  7. Q7Show that $\cos^{-1}\left(\dfrac{4}{5}\right) + \cos^{-1}\left(\dfrac{12}{13}\right) = \cos^{-1}\left(\dfrac{33}{65}\right)$Preview
  8. Q8The principal solutions of $\sec x = \dfrac{2}{\sqrt{3}}$ are ______. (a) $\dfrac{\pi}{3}, \dfrac{11\pi}{6}$ (b) $\dfrac{\pi}{6}, \dfrac{11\…Preview
  9. Q9In $\triangle ABC$, prove that $\tan\left(\dfrac{C-A}{2}\right) = \left(\dfrac{c-a}{c+a}\right)\cot\dfrac{B}{2}$.Preview
  10. Q10Prove that: $\sin^{-1}\left(\dfrac{3}{5}\right) + \cos^{-1}\left(\dfrac{12}{13}\right) = \sin^{-1}\left(\dfrac{56}{65}\right)$.Preview
  11. Q11The principal solutions of $\cot x = -\sqrt{3}$ are ________. (a) $\dfrac{\pi}{6}, \dfrac{5\pi}{6}$ (b) $\dfrac{5\pi}{6}, \dfrac{7\pi}{6}$ (…Preview
  12. Q12In $\triangle ABC$, prove that $\sin\left(\dfrac{B-C}{2}\right) = \left(\dfrac{b-c}{a}\right)\cos\left(\dfrac{A}{2}\right)$ **OR** Show that…Preview
  13. Q13In $\triangle ABC$, with usual notations prove that $b^2 = c^2 + a^2 - 2ca\cos B$ **OR** In $\triangle ABC$, with usual notations prove that…Preview
  14. Q14In $\triangle ABC$, if $a = 2$, $b = 3$ and $\sin A = \dfrac{2}{3}$, then $\angle B = $ ________. (a) $\dfrac{\pi}{4}$ (b) $\dfrac{\pi}{2}$…Preview
  15. Q15Find the general solution of $\tan 2x = 0$Preview
  16. Q16Show that: $\sin^{-1}\left(\dfrac{8}{17}\right) + \sin^{-1}\left(\dfrac{3}{5}\right) = \sin^{-1}\left(\dfrac{77}{85}\right)$.Preview
  17. Q17In $\triangle ABC$, if $a+b+c=2s$, then prove that $\sin\left(\dfrac{A}{2}\right)=\sqrt{\dfrac{(s-b)(s-c)}{bc}}$, with usual notations.Preview
  18. Q18In $\triangle ABC$ if $c^2+a^2-b^2=ac$, then $\angle B = $ ________. (a) $\dfrac{\pi}{4}$ (b) $\dfrac{\pi}{3}$ (c) $\dfrac{\pi}{2}$ (d) $\df…Preview
  19. Q19Find the principal value of $\cot^{-1}\left(\dfrac{-1}{\sqrt 3}\right)$.Preview
  20. Q20Find the principal solutions of $\cot \theta = 0$Preview
  21. Q21Find the cartesian co-ordinates of the point whose polar co-ordinates are $\left(\dfrac{1}{2}, \dfrac{\pi}{3}\right)$.Preview
  22. Q22If $2\tan^{-1}(\cos x) = \tan^{-1}(2\csc x)$, then find the value of $x$.Preview
  23. Q23In $\triangle ABC$, if $c^2+a^2-b^2=ac$, then $\angle B = $ ________. (a) $\dfrac{\pi}{4}$ (b) $\dfrac{\pi}{3}$ (c) $\dfrac{\pi}{2}$ (d) $\d…Preview
  24. Q24Find the cartesian co-ordinates of the point whose polar co-ordinates are $\left(\sqrt 2, \dfrac{\pi}{4}\right)$.Preview
  25. Q25Find the general solution of $\sin\theta + \sin 3\theta + \sin 5\theta = 0$Preview
  26. Q26If $-1 \le x \le 1$, then prove that $\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}$Preview
  27. Q27The principle solutions of the equation $\cos\theta = \dfrac{1}{2}$ are ____. (a) $\dfrac{\pi}{6}, \dfrac{5\pi}{6}$ (b) $\dfrac{\pi}{3}, \df…Preview
  28. Q28In $\triangle ABC$, if $a=18, b=24$ and $c=30$ then find the value of $\sin\left(\dfrac{A}{2}\right)$.Preview
  29. Q29Prove that: $\tan^{-1}\left(\dfrac{1}{2}\right)+\tan^{-1}\left(\dfrac{1}{3}\right)=\dfrac{\pi}{4}$Preview
  30. Q30In $\triangle ABC$, prove that: $\dfrac{\cos A}{a}+\dfrac{\cos B}{b}+\dfrac{\cos C}{c}=\dfrac{a^2+b^2+c^2}{2abc}$.Preview
  31. Q31In $\triangle ABC$, $(a+b)\cdot\cos C+(b+c)\cos A+(c+a)\cdot\cos B$ is equal to ____. (a) $a-b+c$ (b) $a+b-c$ (c) $a+b+c$ (d) $a-b-c$Preview
  32. Q32Find the general solution of $\tan^2\theta=1$.Preview
  33. Q33Prove that: $2\tan^{-1}\left(\dfrac13\right)+\cos^{-1}\left(\dfrac35\right)=\dfrac{\pi}{2}$.Preview
  34. Q34In $\triangle ABC$ if $a=13$, $b=14$, $c=15$ then find the values of (i) $\sec A$ (ii) $\csc \dfrac{A}{2}$.Preview
  35. Q35If $\tan^{-1}(2x)+\tan^{-1}(3x)=\dfrac{\pi}{4}$, then $x=$ ____. (a) -1 (b) $\dfrac16$ (c) $\dfrac13$ (d) $\dfrac32$Preview
  36. Q36Evaluate: $\cos^{-1}\left(\dfrac12\right)+2\sin^{-1}\left(\dfrac12\right)$Preview
  37. Q37In $\triangle ABC$, prove that $a(b\cos C - c\cos B)=b^2-c^2$.Preview
  38. Q38Find the general solution of $4\cos^2\theta=3$.Preview
  39. Q39In $\triangle ABC$, if $A=45°$, $B=60°$ then find the ratio of its sides.Preview

More questions

+Show 73 questions73 questions
  1. Q66The principal solutions of equation $\sin\theta = -\dfrac{1}{2}$ are ____________. (a) $\dfrac{\pi}{6}, \dfrac{5\pi}{6}$ (b) $\dfrac{7\pi}{6…Free
  2. Q67The principal solutions of equation $\cot\theta = \sqrt{3}$ are ____________. (a) $\dfrac{\pi}{6}, \dfrac{7\pi}{6}$ (b) $\dfrac{\pi}{6}, \df…Free
  3. Q68The general solution of $\sec x = \sqrt{2}$ is __________. (a) $2n\pi \pm \dfrac{\pi}{4}$, $n \in Z$ (b) $2n\pi \pm \dfrac{\pi}{2}$, $n \in…Free
  4. Q69If $\cos p\theta = \cos q\theta$, $p \ne q$ then ________. (a) $\theta = \dfrac{2n\pi}{p \pm q}$ (b) $\theta = 2n\pi$ (c) $\theta = 2n\pi +…Preview
  5. Q70If polar co-ordinates of a point are $\left(2, \dfrac{\pi}{4}\right)$ then its Cartesian co-ordinates are ______. (a) $(2, \sqrt{2})$ (b) $(…Preview
  6. Q71If $\sqrt{3}\cos x - \sin x = 1$, then the general value of $x$ is _________. (a) $2n\pi \pm \dfrac{\pi}{3}$ (b) $2n\pi \pm \dfrac{\pi}{6}$…Preview
  7. Q72In $\triangle ABC$ if $\angle A = 45^\circ$, $\angle B = 30^\circ$ then $a:b:c =$ _________. (a) $2 : \sqrt{2} : \sqrt{3}+1$ (b) $\sqrt{2} :…Preview
  8. Q73In $\triangle ABC$, if $c^2 + a^2 - b^2 = ac$, then $\angle B =$ __________. (a) $\dfrac{\pi}{4}$ (b) $\dfrac{\pi}{3}$ (c) $\dfrac{\pi}{2}$…Preview
  9. Q74In $\triangle ABC$, $ac\cos B - bc\cos A =$ ____________. (a) $a^2-b^2$ (b) $b^2-c^2$ (c) $c^2-a^2$ (d) $a^2-b^2-c^2$Preview
  10. Q75If in a triangle, the angles are in A.P. and $b : c = \sqrt{3} : \sqrt{2}$ then $A$ is equal to __________. (a) $30^\circ$ (b) $60^\circ$ (c…Preview
  11. Q76$\cos^{-1}\left(\cos\dfrac{7\pi}{6}\right) =$ ___________. (a) $\dfrac{7\pi}{6}$ (b) $\dfrac{5\pi}{6}$ (c) $\dfrac{\pi}{6}$ (d) $\dfrac{3\pi…Preview
  12. Q77The principal value of $\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$ is ____________. (a) $-\dfrac{2\pi}{3}$ (b) $\dfrac{4\pi}{3}$ (c) $\dfra…Preview
  13. Q78If $\sin^{-1}\dfrac{4}{5} + \cos^{-1}\dfrac{12}{13} = \sin^{-1}a$, then $a =$ _____________. (a) $\dfrac{63}{65}$ (b) $\dfrac{62}{65}$ (c) $…Preview
  14. Q79If $\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}$, then $x =$ (a) $-1$ (b) $\dfrac{1}{6}$ (c) $\dfrac{2}{6}$ (d) $\dfrac{3}{2}$Preview
  15. Q80$2\tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{7} =$ __________. (a) $\tan^{-1}\dfrac{4}{5}$ (b) $\dfrac{\pi}{2}$ (c) $1$ (d) $\dfrac{\pi}{4}$Preview
  16. Q81$\tan\left(\dfrac{\pi}{4} - 2\tan^{-1}\dfrac{1}{5}\right) =$ __________. (a) $\dfrac{7}{17}$ (b) $\dfrac{17}{7}$ (c) $-\dfrac{7}{17}$ (d) $-…Preview
  17. Q82The principal value branch of $\sec^{-1}x$ is __________. (a) $(0, \pi)$ (b) $[0, \pi]$ (c) $[0,\pi] - \left\{\dfrac{\pi}{2}\right\}$ (d) $\…Preview
  18. Q83$\cos\left(\tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{2}\right) =$ ____________. (a) $\dfrac{1}{\sqrt{2}}$ (b) $\dfrac{\sqrt{3}}{2}$ (c) $\d…Preview
  19. Q84If $\tan\theta + \tan 2\theta + \tan 3\theta = \tan\theta\tan 2\theta\tan 3\theta$, then the general value of $\theta$ is _______. (a) $n\pi…Preview
  20. Q85In any $\triangle ABC$, if $a\cos B = b\cos A$, then the triangle is ________. (a) Equilateral triangle (b) Isosceles triangle (c) Scalene (…Preview
  21. Q86Find the principal solutions of the equation $\sin 2\theta = -\dfrac{1}{2}$Preview
  22. Q87Find the principal solutions of the equation $\tan 3\theta = -1$Preview
  23. Q88Find the principal solutions of the equation $\cot\theta = 0$Preview
  24. Q89Find the principal solutions of the equation $\sin 2\theta = -\dfrac{1}{2}$Preview
  25. Q90Find the principal solutions of the equation $\tan 5\theta = -1$Preview
  26. Q91Find the principal solutions of the equation $\cot 2\theta = 0$Preview
  27. Q92Which of the following equations have no solutions? (i) $\cos 2\theta = \dfrac{1}{3}$ (ii) $\cos^2\theta = -1$ (iii) $2\sin\theta = 3$ (iv)…Preview
  28. Q93Find the general solution of the equation $\tan\theta = -\sqrt{3}$Preview
  29. Q94Find the general solution of the equation $\tan^2\theta = 3$Preview
  30. Q95Find the general solution of the equation $\sin\theta - \cos\theta = 1$Preview
  31. Q96Find the general solution of the equation $\sin 2\theta - \cos 2\theta = 1$Preview
  32. Q97In $\triangle ABC$ prove that $\dfrac{\cos\dfrac{A-B}{2}}{\sin\dfrac{C}{2}} = \dfrac{a+b}{c}$.Preview
  33. Q98With usual notations prove that $(a-b)^2\cos^2\dfrac{C}{2} + (a+b)^2\sin^2\dfrac{C}{2} = c^2$.Preview
  34. Q99In $\triangle ABC$ prove that $(a+b)^2\cos^2\dfrac{C}{2} + (a-b)^2\sin^2\dfrac{C}{2} = a^2 + b^2 + 2ab\cos C$.Preview
  35. Q100In $\triangle ABC$ if $\cos A = \sin B - \cos C$ then show that it is a right angled triangle.Preview
  36. Q101[Source text badly corrupted at this item — the printed condition involving $\sin(A-B)$, $\sin(B-?)$ and side $c$ could not be reliably reco…Preview
  37. Q102Solve the triangle in which $a = \sqrt{3}+1$, $b = \sqrt{3}-1$ and $C = 60^\circ$.Preview
  38. Q103In $\triangle ABC$ prove that $a\sin A - b\sin B = c\sin(A-B)$.Preview
  39. Q104In $\triangle ABC$ prove that $\dfrac{c - b\cos A}{b - c\cos A} = \dfrac{\cos B}{\cos C}$.Preview
  40. Q105In $\triangle ABC$ prove that $a^2\sin(B-C) = (b^2-c^2)\sin A$.Preview
  41. Q106In $\triangle ABC$ prove that $ac\cos B - bc\cos A = a^2 - b^2$.Preview
  42. Q107In $\triangle ABC$ prove that $\dfrac{\cos A}{a} + \dfrac{\cos B}{b} + \dfrac{\cos C}{c} = \dfrac{a^2+b^2+c^2}{2abc}$.Preview
  43. Q108In $\triangle ABC$ prove that $\dfrac{1-\cos 2A}{a^2} = \dfrac{1-\cos 2B}{b^2}$.Preview
  44. Q109In $\triangle ABC$ prove that $\dfrac{\tan\dfrac{B-C}{2}}{\tan\dfrac{B+C}{2}} = \dfrac{b-c}{b+c}$.Preview
  45. Q110In $\triangle ABC$ if $a^2$, $b^2$, $c^2$ are in A.P. then show that $\cot A$, $\cot B$, $\cot C$ are also in A.P.Preview
  46. Q111In $\triangle ABC$ if $C = 90^\circ$ then prove that $\sin(A-B) = \dfrac{a^2-b^2}{a^2+b^2}$.Preview
  47. Q112In $\triangle ABC$ if $\dfrac{\cos A}{a} = \dfrac{\cos B}{b}$ then show that it is an isosceles triangle.Preview
  48. Q113In $\triangle ABC$ if $\sin^2 A + \sin^2 B = \sin^2 C$ then prove that the triangle is a right angled triangle.Preview
  49. Q114In $\triangle ABC$ prove that $a^2(\cos^2 B - \cos^2 C) + b^2(\cos^2 C - \cos^2 A) + c^2(\cos^2 A - \cos^2 B) = 0$.Preview
  50. Q115With usual notations show that $(c^2-a^2+b^2)\tan A = (a^2-b^2+c^2)\tan B = (b^2-c^2+a^2)\tan C$.Preview
  51. Q116In $\triangle ABC$ if $b\cos^2\dfrac{A}{2} + a\cos^2\dfrac{B}{2} = \dfrac{3c}{2}$ then prove that $a$, $b$, $c$ are in A.P.Preview
  52. Q117Show that $2\sin^{-1}\dfrac{3}{5} = \tan^{-1}\dfrac{24}{7}$.Preview
  53. Q118Show that $\tan^{-1}\dfrac{1}{5} + \tan^{-1}\dfrac{1}{7} + \tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{8} = \dfrac{\pi}{4}$.Preview
  54. Q119Prove that $\tan^{-1}x = \dfrac{1}{2}\cos^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)$, if $x \in [0,1]$.Preview
  55. Q120[Source text badly corrupted at this item — the numeric constants of this $\sin^{-1}$ identity could not be reliably reconstructed from the…Preview
  56. Q121Show that $\tan^{-1}\sqrt{\dfrac{1-x}{1+x}} = \dfrac{\pi}{4} - \dfrac{1}{2}\cos^{-1}x$, for $-1 \le x \le 1$.Preview
  57. Q122If $\sin\left(\sin^{-1}\dfrac{1}{5} + \cos^{-1}x\right) = 1$ then find the value of $x$.Preview
  58. Q123If $\tan^{-1}\dfrac{x-1}{x-2} + \tan^{-1}\dfrac{x+1}{x+2} = \dfrac{\pi}{4}$ then find the value of $x$.Preview
  59. Q124If $2\tan^{-1}(\cos x) = \tan^{-1}(\text{cosec}\,x)$ then find the value of $x$.Preview
  60. Q125Solve: $\tan^{-1}\dfrac{1-x}{1+x} = \dfrac{1}{2}\tan^{-1}x$, for $x > 0$.Preview
  61. Q126If $\sin^{-1}(1-x) - 2\sin^{-1}x = \dfrac{\pi}{2}$ then find the value of $x$.Preview
  62. Q127If $\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{2}$ then find the value of $x$.Preview
  63. Q128Show that $\tan^{-1}\dfrac{1}{2} + \tan^{-1}\dfrac{1}{5} + \tan^{-1}\dfrac{1}{8} = \dfrac{\pi}{4}$.Preview
  64. Q129Show that $\cot^{-1}\dfrac{1}{3} - \tan^{-1}\dfrac{1}{3} = \cot^{-1}\dfrac{3}{4}$.Preview
  65. Q130[Source text badly corrupted at this item — the numeric constants of this $\tan^{-1}$ identity could not be reliably reconstructed from the…Preview
  66. Q131[Source text badly corrupted at this item — the numeric constants of this inverse-trigonometric identity could not be reliably reconstructed…Preview
  67. Q132[Source text badly corrupted at this item — the numeric constants of this $\cot^{-1}$/$\sec^{-1}$ identity could not be reliably reconstruct…Preview
  68. Q133Prove that $\cos^{-1}x = 2\tan^{-1}\sqrt{\dfrac{1-x}{1+x}}$, for $-1 \le x \le 1$.Preview
  69. Q134[Source text badly corrupted at this item — the second sub-identity in this pair could not be reliably distinguished from the first during e…Preview
  70. Q135If $|x| < 1$, then prove that $2\tan^{-1}x = \tan^{-1}\dfrac{2x}{1-x^2} = \sin^{-1}\dfrac{2x}{1+x^2} = \cos^{-1}\dfrac{1-x^2}{1+x^2}$.Preview
  71. Q136If $x, y, z$ are positive then prove that $\tan^{-1}\dfrac{x-y}{1+xy} + \tan^{-1}\dfrac{y-z}{1+yz} + \tan^{-1}\dfrac{z-x}{1+zx} = 0$.Preview
  72. Q137If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \dfrac{\pi}{2}$ then show that $xy + yz + zx = 1$.Preview
  73. Q138If $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi$ then show that $x^2 + y^2 + z^2 + 2xyz = 1$.Preview