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Mathematics · Class 11 Science

Ch 3Trigonometric Functions — Class 11 Mathematics, concept-first.

The word trigonometry comes from two Greek words: trigon (meaning a triangle) and metron (meaning to measure). So, at its root, trigonometry is the study of measuring triangles.

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Key concepts

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Angle Conversion

Imagine you're measuring the length of a table. You could use centimetres, inches, or feet — all are valid, but the number changes depending on which unit you pick. The same idea applies to angles.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

3.1

Introduction

The word trigonometry comes from two Greek words: trigon (meaning a triangle) and metron (meaning to measure). So, at its root, trigonometry is the study of measuring triangles.

3.2

Angles

An angle is not merely a static shape formed by two rays meeting at a point. In trigonometry, we treat an angle dynamically — as the measure of rotation of a given ray about its fixed endpoint.

3.2.1

Degree Measure

An angle is formed by rotating a ray (the initial side) to a new position (the terminal side). The size of the angle depends on how much rotation has occurred.

3.2.2

Radian Measure

Degrees are not the only way to measure an angle. There is a more natural unit, called the radian, that is based directly on the geometry of a circle. The definition is simple and powerful.

3.2.3

Relation Between Radian and Real Numbers

The connection between radian measure and real numbers is not a theorem to be memorised — it is a definition that makes trigonometry work as a function of real numbers.

3.2.4

Relation Between Degree and Radian

12 Q

The entire bridge between degree measure and radian measure rests on one simple observation: a full circle is simultaneously 360° and radians.

+Worked Examplesi5 questions
  1. Example 1Convert $40^\circ\, 20'$ into radian measure.Free
  2. Example 2Convert $6$ radians into degree measure.Free
  3. Example 3Find the radius of the circle in which a central angle of $60^\circ$ intercepts an arc of length $37.4$ cm (use $\pi = \frac{22}{7}$).Preview
  4. Example 4The minute hand of a watch is $1.5$ cm long. How far does its tip move in $40$ minutes? (Use $\pi = 3.14$.)Preview
  5. Example 5If the arcs of the same lengths in two circles subtend angles $65^\circ$ and $110^\circ$ at the centre, find the ratio of their radii.Preview
+Exercise 3.1i7 questions
  1. Q1Find the radian measures corresponding to the following degree measures: (i) $25^\circ$ (ii) $-47^\circ 30'$ (iii) $240^\circ$ (iv) $520^\ci…Free
  2. Q2Find the degree measures corresponding to the following radian measures (Use $\pi = \frac{22}{7}$). (i) $\frac{11}{16}$ (ii) $-4$ (iii) $\fr…Free
  3. Q3A wheel makes $360$ revolutions in one minute. Through how many radians does it turn in one second?Free
  4. Q4Find 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 $\pi = \frac{22…Preview
  5. Q5In a circle of diameter $40$ cm, the length of a chord is $20$ cm. Find the length of minor arc of the chord.Preview
  6. Q6If in two circles, arcs of the same length subtend angles $60^\circ$ and $75^\circ$ at the centre, find the ratio of their radii.Preview
  7. Q7Find the angle in radian through which a pendulum swings if its length is $75$ cm and the tip describes an arc of length (i) $10$ cm (ii) $1…Preview
3.3

Trigonometric Functions

In earlier classes, you studied trigonometric ratios for acute angles — those were ratios of sides in a right-angled triangle. Now we take a fundamentally different approach.

3.3.1

Sign of Trigonometric Functions

The sign of a trigonometric function for a given angle is determined entirely by the quadrant in which the terminal side of the angle lies. To see why, we return to the unit circle definition.

3.3.2

Domain and Range of Trigonometric Functions

14 Q

The sine and cosine functions are defined for every real number. If you take any real , you can find and — there is no restriction. That is the first observation.

+Worked Examplesi4 questions
  1. Example 6If $\cos x = -\frac{3}{5}$, $x$ lies in the third quadrant, find the values of other five trigonometric functions.Free
  2. Example 7If $\cot x = -\frac{5}{12}$, $x$ lies in second quadrant, find the values of other five trigonometric functions.Free
  3. Example 8Find the value of $\sin\frac{31\pi}{3}$.Preview
  4. Example 9Find the value of $\cos(-1710^\circ)$.Preview
+Exercise 3.2i10 questions
  1. Q1Find the values of other five trigonometric functions if $\cos x = -\frac{1}{2}$, $x$ lies in third quadrant.Free
  2. Q2Find the values of other five trigonometric functions if $\sin x = \frac{3}{5}$, $x$ lies in second quadrant.Free
  3. Q3Find the values of other five trigonometric functions if $\cot x = \frac{4}{3}$, $x$ lies in third quadrant.Free
  4. Q4Find the values of other five trigonometric functions if $\sec x = \frac{13}{5}$, $x$ lies in fourth quadrant.Preview
  5. Q5Find the values of other five trigonometric functions if $\tan x = -\frac{5}{12}$, $x$ lies in second quadrant.Preview
  6. Q6Find the value of the trigonometric function $\sin 765^\circ$.Preview
  7. Q7Find the value of the trigonometric function $\csc(-1410^\circ)$.Preview
  8. Q8Find the value of the trigonometric function $\tan\frac{19\pi}{3}$.Preview
  9. Q9Find the value of the trigonometric function $\sin\left(-\frac{11\pi}{3}\right)$.Preview
  10. Q10Find the value of the trigonometric function $\cot\left(-\frac{15\pi}{4}\right)$.Preview
3.4

Trigonometric Functions of Sum and Difference of Two Angles

33 Q

This section develops the core identities that let you express trigonometric functions of sums and differences in terms of functions of the individual angles.

+Worked Examplesi8 questions
  1. Example 10Prove that $3\sin\frac{\pi}{6}\sec\frac{\pi}{3} - 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} = 1$.Free
  2. Example 11Find the value of $\sin 15^\circ$.Free
  3. Example 12Find the value of $\tan\frac{13\pi}{12}$.Free
  4. Example 13Prove that $\dfrac{\sin(x+y)}{\sin(x-y)} = \dfrac{\tan x + \tan y}{\tan x - \tan y}$.Preview
  5. Example 14Show that $\tan 3x\, \tan 2x\, \tan x = \tan 3x - \tan 2x - \tan x$.Preview
  6. Example 15Prove that $\cos\left(\frac{\pi}{4}+x\right) + \cos\left(\frac{\pi}{4}-x\right) = \sqrt{2}\,\cos x$.Preview
  7. Example 16Prove that $\dfrac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \cot x$.Preview
  8. Example 17Prove that $\dfrac{\sin 5x - 2\sin 3x + \sin x}{\cos 5x - \cos x} = \tan x$.Preview
+Exercise 3.3i25 questions
  1. Q1Prove that $\sin^2\frac{\pi}{6} + \cos^2\frac{\pi}{3} - \tan^2\frac{\pi}{4} = -\frac{1}{2}$.Free
  2. Q2Prove that $2\sin^2\frac{\pi}{6} + \csc^2\frac{7\pi}{6}\cos^2\frac{\pi}{3} = \frac{3}{2}$.Free
  3. Q3Prove that $\cot^2\frac{\pi}{6} + \csc\frac{5\pi}{6} + 3\tan^2\frac{\pi}{6} = 6$.Free
  4. Q4Prove that $2\sin^2\frac{3\pi}{4} + 2\cos^2\frac{\pi}{4} + 2\sec^2\frac{\pi}{3} = 10$.Preview
  5. Q5Find the value of: (i) $\sin 75^\circ$ (ii) $\tan 15^\circ$Preview
  6. Q6Prove that $\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\rig…Preview
  7. Q7Prove that $\dfrac{\tan\left(\frac{\pi}{4}+x\right)}{\tan\left(\frac{\pi}{4}-x\right)} = \left(\dfrac{1+\tan x}{1-\tan x}\right)^2$.Preview
  8. Q8Prove that $\dfrac{\cos(\pi+x)\cos(-x)}{\sin(\pi-x)\cos\left(\frac{\pi}{2}+x\right)} = \cot^2 x$.Preview
  9. Q9Prove that $\cos\left(\frac{3\pi}{2}+x\right)\cos(2\pi+x)\left[\cot\left(\frac{3\pi}{2}-x\right) + \cot(2\pi+x)\right] = 1$.Preview
  10. Q10Prove that $\sin(n+1)x\, \sin(n+2)x + \cos(n+1)x\, \cos(n+2)x = \cos x$.Preview
  11. Q11Prove that $\cos\left(\frac{3\pi}{4}+x\right) - \cos\left(\frac{3\pi}{4}-x\right) = -\sqrt{2}\,\sin x$.Preview
  12. Q12Prove that $\sin^2 6x - \sin^2 4x = \sin 2x\, \sin 10x$.Preview
  13. Q13Prove that $\cos^2 2x - \cos^2 6x = \sin 4x\, \sin 8x$.Preview
  14. Q14Prove that $\sin 2x + 2\sin 4x + \sin 6x = 4\cos^2 x\, \sin 4x$.Preview
  15. Q15Prove that $\cot 4x\,(\sin 5x + \sin 3x) = \cot x\,(\sin 5x - \sin 3x)$.Preview
  16. Q16Prove that $\dfrac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\dfrac{\sin 2x}{\cos 10x}$.Preview
  17. Q17Prove that $\dfrac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x$.Preview
  18. Q18Prove that $\dfrac{\sin x - \sin y}{\cos x + \cos y} = \tan\frac{x-y}{2}$.Preview
  19. Q19Prove that $\dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x$.Preview
  20. Q20Prove that $\dfrac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2\sin x$.Preview
  21. Q21Prove that $\dfrac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x$.Preview
  22. Q22Prove that $\cot x\, \cot 2x - \cot 2x\, \cot 3x - \cot 3x\, \cot x = 1$.Preview
  23. Q23Prove that $\tan 4x = \dfrac{4\tan x\,(1 - \tan^2 x)}{1 - 6\tan^2 x + \tan^4 x}$.Preview
  24. Q24Prove that $\cos 4x = 1 - 8\sin^2 x\, \cos^2 x$.Preview
  25. Q25Prove that $\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1$.Preview

Miscellaneous Examples

+Miscellaneous Examplesi5 questions
  1. Example 18If $\sin x = \frac{3}{5}$, $\cos y = -\frac{12}{13}$, where $x$ and $y$ both lie in second quadrant, find the value of $\sin(x + y)$.Free
  2. Example 19Prove that $\cos 2x\, \cos\frac{x}{2} - \cos 3x\, \cos\frac{9x}{2} = \sin 5x\, \sin\frac{5x}{2}$.Free
  3. Example 20Find the value of $\tan\frac{\pi}{8}$.Preview
  4. Example 21If $\tan x = \frac{3}{4}$, $\pi < x < \frac{3\pi}{2}$, find the value of $\sin\frac{x}{2}$, $\cos\frac{x}{2}$ and $\tan\frac{x}{2}$.Preview
  5. Example 22Prove that $\cos^2 x + \cos^2\left(x + \frac{\pi}{3}\right) + \cos^2\left(x - \frac{\pi}{3}\right) = \frac{3}{2}$.Preview

Miscellaneous Exercise on Chapter 3

+Miscellaneous Exercisei10 questions
  1. Q1Prove that $2\cos\frac{\pi}{13}\cos\frac{9\pi}{13} + \cos\frac{3\pi}{13} + \cos\frac{5\pi}{13} = 0$.Free
  2. Q2Prove that $(\sin 3x + \sin x)\sin x + (\cos 3x - \cos x)\cos x = 0$.Free
  3. Q3Prove that $(\cos x + \cos y)^2 + (\sin x - \sin y)^2 = 4\cos^2\frac{x+y}{2}$.Free
  4. Q4Prove that $(\cos x - \cos y)^2 + (\sin x - \sin y)^2 = 4\sin^2\frac{x-y}{2}$.Preview
  5. Q5Prove that $\sin x + \sin 3x + \sin 5x + \sin 7x = 4\cos x\, \cos 2x\, \sin 4x$.Preview
  6. Q6Prove that $\dfrac{(\sin 7x + \sin 5x) + (\sin 9x + \sin 3x)}{(\cos 7x + \cos 5x) + (\cos 9x + \cos 3x)} = \tan 6x$.Preview
  7. Q7Prove that $\sin 3x + \sin 2x - \sin x = 4\sin x\, \cos\frac{x}{2}\, \cos\frac{3x}{2}$.Preview
  8. Q8Find $\sin\frac{x}{2}$, $\cos\frac{x}{2}$ and $\tan\frac{x}{2}$ if $\tan x = -\frac{4}{3}$, $x$ in quadrant II.Preview
  9. Q9Find $\sin\frac{x}{2}$, $\cos\frac{x}{2}$ and $\tan\frac{x}{2}$ if $\cos x = -\frac{1}{3}$, $x$ in quadrant III.Preview
  10. Q10Find $\sin\frac{x}{2}$, $\cos\frac{x}{2}$ and $\tan\frac{x}{2}$ if $\sin x = \frac{1}{4}$, $x$ in quadrant II.Preview

Summary

- Angle measurement: Radian measure is standard; . Arc length , area of sector . - Trigonometric ratios: For an angle in standard position, , , (), with reciprocals , , .

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 76 questions76 questions
  1. Q1Prove that $\dfrac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \dfrac{1 + \sin A}{\cos A}$.Free
  2. Q2If $\dfrac{2\sin\alpha}{1 + \cos\alpha + \sin\alpha} = y$, then prove that $\dfrac{1 - \cos\alpha + \sin\alpha}{1 + \sin\alpha}$ is also equ…Free
  3. Q3If $m\sin\theta = n\sin(\theta + 2\alpha)$, then prove that $\tan(\theta + \alpha)\cot\alpha = \dfrac{m + n}{m - n}$.Free
  4. Q4If $\cos(\alpha + \beta) = \dfrac{4}{5}$ and $\sin(\alpha - \beta) = \dfrac{5}{13}$, where $\alpha$ lie between $0$ and $\dfrac{\pi}{4}$, fi…Preview
  5. Q5If $\tan x = \dfrac{b}{a}$, then find the value of $\sqrt{\dfrac{a + b}{a - b}} + \sqrt{\dfrac{a - b}{a + b}}$.Preview
  6. Q6Prove that $\cos\theta\,\cos\dfrac{\theta}{2} - \cos 3\theta\,\cos\dfrac{9\theta}{2} = \sin\dfrac{7\theta}{2}\,\sin 4\theta$.Preview
  7. Q7If $a\cos\theta + b\sin\theta = m$ and $a\sin\theta - b\cos\theta = n$, then show that $a^2 + b^2 = m^2 + n^2$.Preview
  8. Q8Find the value of $\tan 22^\circ 30'$.Preview
  9. Q9Prove that $\sin 4A = 4\sin A\cos^3 A - 4\cos A\sin^3 A$.Preview
  10. Q10If $\tan\theta + \sin\theta = m$ and $\tan\theta - \sin\theta = n$, then prove that $m^2 - n^2 = 4\sin\theta\,\tan\theta$.Preview
  11. Q11If $\tan(A + B) = p$, $\tan(A - B) = q$, then show that $\tan 2A = \dfrac{p + q}{1 - pq}$.Preview
  12. Q12If $\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta$, then prove that $\cos 2\alpha + \cos 2\beta = -2\cos(\alpha + \beta)$.Preview
  13. Q13If $\dfrac{\sin(x + y)}{\sin(x - y)} = \dfrac{a + b}{a - b}$, then show that $\dfrac{\tan x}{\tan y} = \dfrac{a}{b}$.Preview
  14. Q14If $\tan\theta = \dfrac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}$, then show that $\sin\alpha + \cos\alpha = \sqrt{2}\,\cos\theta$.Preview
  15. Q15If $\sin\theta + \cos\theta = 1$, then find the general value of $\theta$.Preview
  16. Q16Find the most general value of $\theta$ satisfying the equation $\tan\theta = -1$ and $\cos\theta = \dfrac{1}{\sqrt{2}}$.Preview
  17. Q17If $\cot\theta + \tan\theta = 2\csc\theta$, then find the general value of $\theta$.Preview
  18. Q18If $2\sin^2\theta = 3\cos\theta$, where $0 \le \theta \le 2\pi$, then find the value of $\theta$.Preview
  19. Q19If $\sec x\,\cos 5x + 1 = 0$, where $0 < x \le \dfrac{\pi}{2}$, then find the value of $x$.Preview
  20. Q20If $\sin(\theta + \alpha) = a$ and $\sin(\theta + \beta) = b$, then prove that $\cos 2(\alpha - \beta) - 4ab\cos(\alpha - \beta) = 1 - 2a^2…Preview
  21. Q21If $\cos(\theta + \phi) = m\cos(\theta - \phi)$, then prove that $\tan\theta = \dfrac{1 - m}{1 + m}\cot\phi$.Preview
  22. Q22Find the value of the expression $3\left[\sin^4\left(\dfrac{3\pi}{2} - \alpha\right) + \sin^4(3\pi + \alpha)\right] - 2\left[\sin^6\left(\df…Preview
  23. Q23If $a\cos 2\theta + b\sin 2\theta = c$ has $\alpha$ and $\beta$ as its roots, then prove that $\tan\alpha + \tan\beta = \dfrac{2b}{a + c}$.Preview
  24. Q24If $x = \sec\phi - \tan\phi$ and $y = \csc\phi + \cot\phi$ then show that $xy + x - y + 1 = 0$.Preview
  25. Q25If $\theta$ lies in the first quadrant and $\cos\theta = \dfrac{8}{17}$, then find the value of $\cos(30^\circ + \theta) + \cos(45^\circ - \…Preview
  26. Q26Find the value of the expression $\cos^4\dfrac{\pi}{8} + \cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4\dfrac{7\pi}{8}$.Preview
  27. Q27Find the general solution of the equation $5\cos^2\theta + 7\sin^2\theta - 6 = 0$.Preview
  28. Q28Find the general solution of the equation $\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$.Preview
  29. Q29Find the general solution of the equation $(\sqrt{3} - 1)\cos\theta + (\sqrt{3} + 1)\sin\theta = 2$.Preview
  30. Q30If $\sin\theta + \csc\theta = 2$, then $\sin^2\theta + \csc^2\theta$ is equal to (A) $1$ (B) $4$ (C) $2$ (D) None of thesePreview
  31. Q31If $f(x) = \cos^2 x + \sec^2 x$, then (A) $f(x) < 1$ (B) $f(x) = 1$ (C) $1 < f(x) < 2$ (D) $f(x) \ge 2$Preview
  32. Q32If $\tan\theta = \dfrac{1}{2}$ and $\tan\phi = \dfrac{1}{3}$, then the value of $\theta + \phi$ is (A) $\dfrac{\pi}{6}$ (B) $\pi$ (C) $0$ (D…Preview
  33. Q33Which of the following is not correct? (A) $\sin\theta = -\dfrac{1}{5}$ (B) $\cos\theta = 1$ (C) $\sec\theta = \dfrac{1}{2}$ (D) $\tan\theta…Preview
  34. Q34The value of $\tan 1^\circ\,\tan 2^\circ\,\tan 3^\circ \ldots \tan 89^\circ$ is (A) $0$ (B) $1$ (C) $\dfrac{1}{2}$ (D) Not definedPreview
  35. Q35The value of $\dfrac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ}$ is (A) $1$ (B) $\sqrt{3}$ (C) $\dfrac{\sqrt{3}}{2}$ (D) $2$Preview
  36. Q36The value of $\cos 1^\circ\,\cos 2^\circ\,\cos 3^\circ \ldots \cos 179^\circ$ is (A) $\dfrac{1}{\sqrt{2}}$ (B) $0$ (C) $1$ (D) $-1$Preview
  37. Q37If $\tan\theta = 3$ and $\theta$ lies in third quadrant, then the value of $\sin\theta$ is (A) $\dfrac{1}{\sqrt{10}}$ (B) $-\dfrac{1}{\sqrt{…Preview
  38. Q38The value of $\tan 75^\circ - \cot 75^\circ$ is equal to (A) $2\sqrt{3}$ (B) $2 + \sqrt{3}$ (C) $2 - \sqrt{3}$ (D) $1$Preview
  39. Q39Which of the following is correct? (A) $\sin 1^\circ > \sin 1$ (B) $\sin 1^\circ < \sin 1$ (C) $\sin 1^\circ = \sin 1$ (D) $\sin 1^\circ = \…Preview
  40. Q40If $\tan\alpha = \dfrac{m}{m + 1}$, $\tan\beta = \dfrac{1}{2m + 1}$, then $\alpha + \beta$ is equal to (A) $\dfrac{\pi}{2}$ (B) $\dfrac{\pi}…Preview
  41. Q41The minimum value of $3\cos x + 4\sin x + 8$ is (A) $5$ (B) $9$ (C) $7$ (D) $3$Preview
  42. Q42The value of $\tan 3A - \tan 2A - \tan A$ is equal to (A) $\tan 3A\,\tan 2A\,\tan A$ (B) $-\tan 3A\,\tan 2A\,\tan A$ (C) $\tan A\,\tan 2A -…Preview
  43. Q43The value of $\sin(45^\circ + \theta) - \cos(45^\circ - \theta)$ is (A) $2\cos\theta$ (B) $2\sin\theta$ (C) $1$ (D) $0$Preview
  44. Q44The value of $\cot\left(\dfrac{\pi}{4} + \theta\right)\cot\left(\dfrac{\pi}{4} - \theta\right)$ is (A) $-1$ (B) $0$ (C) $1$ (D) Not definedPreview
  45. Q45$\cos 2\theta\,\cos 2\phi + \sin^2(\theta - \phi) - \sin^2(\theta + \phi)$ is equal to (A) $\sin 2(\theta + \phi)$ (B) $\cos 2(\theta + \phi…Preview
  46. Q46The value of $\cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ$ is (A) $\dfrac{1}{2}$ (B) $1$ (C) $-\dfrac{1}{2}$ (D) $\dfrac…Preview
  47. Q47If $\tan A = \dfrac{1}{2}$, $\tan B = \dfrac{1}{3}$, then $\tan(2A + B)$ is equal to (A) $1$ (B) $2$ (C) $3$ (D) $4$Preview
  48. Q48The value of $\sin\dfrac{\pi}{10}\,\sin\dfrac{13\pi}{10}$ is (A) $\dfrac{1}{2}$ (B) $-\dfrac{1}{2}$ (C) $-\dfrac{1}{4}$ (D) $1$Preview
  49. Q49The value of $\sin 50^\circ - \sin 70^\circ + \sin 10^\circ$ is equal to (A) $1$ (B) $0$ (C) $\dfrac{1}{2}$ (D) $2$Preview
  50. Q50If $\sin\theta + \cos\theta = 1$, then the value of $\sin 2\theta$ is equal to (A) $1$ (B) $\dfrac{1}{2}$ (C) $0$ (D) $-1$Preview
  51. Q51If $\alpha + \beta = \dfrac{\pi}{4}$, then the value of $(1 + \tan\alpha)(1 + \tan\beta)$ is (A) $1$ (B) $2$ (C) $-2$ (D) Not definedPreview
  52. Q52If $\sin\theta = \dfrac{-4}{5}$ and $\theta$ lies in third quadrant then the value of $\cos\dfrac{\theta}{2}$ is (A) $\dfrac{1}{5}$ (B) $-\d…Preview
  53. Q53Number of solutions of the equation $\tan x + \sec x = 2\cos x$ lying in the interval $[0, 2\pi]$ is (A) $0$ (B) $1$ (C) $2$ (D) $3$Preview
  54. Q54The value of $\sin\dfrac{\pi}{18} + \sin\dfrac{\pi}{9} + \sin\dfrac{2\pi}{9} + \sin\dfrac{5\pi}{18}$ is given by (A) $\sin\dfrac{7\pi}{18} +…Preview
  55. Q55If $A$ lies in the second quadrant and $3\tan A + 4 = 0$, then the value of $2\cot A - 5\cos A + \sin A$ is equal to (A) $\dfrac{-53}{10}$ (…Preview
  56. Q56The value of $\cos^2 48^\circ - \sin^2 12^\circ$ is (A) $\dfrac{\sqrt{5} + 1}{8}$ (B) $\dfrac{\sqrt{5} - 1}{8}$ (C) $\dfrac{\sqrt{5} + 1}{5}…Preview
  57. Q57If $\tan\alpha = \dfrac{1}{7}$, $\tan\beta = \dfrac{1}{3}$, then $\cos 2\alpha$ is equal to (A) $\sin 2\beta$ (B) $\sin 4\beta$ (C) $\sin 3\…Preview
  58. Q58If $\tan\theta = \dfrac{a}{b}$, then $b\cos 2\theta + a\sin 2\theta$ is equal to (A) $a$ (B) $b$ (C) $\dfrac{a}{b}$ (D) NonePreview
  59. Q59If for real values of $x$, $\cos\theta = x + \dfrac{1}{x}$, then (A) $\theta$ is an acute angle (B) $\theta$ is right angle (C) $\theta$ is…Preview
  60. Q60The value of $\dfrac{\sin 50^\circ}{\sin 130^\circ}$ is ______.Preview
  61. Q61If $k = \sin\left(\dfrac{\pi}{18}\right)\sin\left(\dfrac{5\pi}{18}\right)\sin\left(\dfrac{7\pi}{18}\right)$, then the numerical value of $k$…Preview
  62. Q62If $\tan A = \dfrac{1 - \cos B}{\sin B}$, then $\tan 2A = $ ______.Preview
  63. Q63If $\sin x + \cos x = a$, then (i) $\sin^6 x + \cos^6 x = $ ______ (ii) $|\sin x - \cos x| = $ ______.Preview
  64. Q64In a triangle $ABC$ with $\angle C = 90^\circ$ the equation whose roots are $\tan A$ and $\tan B$ is ______.Preview
  65. Q65$3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x) = $ ______.Preview
  66. Q66Given $x > 0$, the values of $f(x) = -3\cos\sqrt{3 + x + x^2}$ lie in the interval ______.Preview
  67. Q67The maximum distance of a point on the graph of the function $y = \sqrt{3}\,\sin x + \cos x$ from $x$-axis is ______.Preview
  68. Q68If $\tan A = \dfrac{1 - \cos B}{\sin B}$, then $\tan 2A = \tan B$.Preview
  69. Q69The equality $\sin A + \sin 2A + \sin 3A = 3$ holds for some real value of $A$.Preview
  70. Q70$\sin 10^\circ$ is greater than $\cos 10^\circ$.Preview
  71. Q71$\cos\dfrac{2\pi}{15}\,\cos\dfrac{4\pi}{15}\,\cos\dfrac{8\pi}{15}\,\cos\dfrac{16\pi}{15} = \dfrac{1}{16}$.Preview
  72. Q72One value of $\theta$ which satisfies the equation $\sin^4\theta - 2\sin^2\theta - 1$ lies between $0$ and $2\pi$.Preview
  73. Q73If $\csc x = 1 + \cot x$ then $x = 2n\pi,\ 2n\pi + \dfrac{\pi}{2}$.Preview
  74. Q74If $\tan\theta + \tan 2\theta + \sqrt{3}\,\tan\theta\,\tan 2\theta = \sqrt{3}$, then $\theta = \dfrac{n\pi}{3} + \dfrac{\pi}{9}$.Preview
  75. Q75If $\tan(\pi\cos\theta) = \cot(\pi\sin\theta)$, then $\cos\left(\theta - \dfrac{\pi}{4}\right) = \pm\dfrac{1}{2\sqrt{2}}$.Preview
  76. Q76Match each item given under the column $C_1$ to its correct answer given under the column $C_2$. Column $C_1$: (a) $\sin(x + y)\sin(x - y)$;…Preview