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Mathematics · Class 11 Science

Ch 2Trigonometry - I — Class 11 Mathematics, concept-first.

Trigonometry studies the relationship between the sides and angles of triangles. The name comes from the Greek words trigonon (triangle) and metron (measure) — literally, 'triangle measurement'.

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Trigonometric Functions of Specific and Allied Angles

Many trigonometric values used throughout mathematics come from a small set of specific angles — 0°, 30°, 45°, 60°, 90°, and angles built from them such as 120°, 225°, and negative angles like −60° — whose exact values c…

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2.1

Introduction

Trigonometry studies the relationship between the sides and angles of triangles. The name comes from the Greek words trigonon (triangle) and metron (measure) — literally, 'triangle measurement'.

+Exercise 2.1i18 questions
  1. Q1Find the trigonometric functions of $0°, 30°, 45°, 60°, 150°, 180°, 210°, 300°, 330°, −30°, −45°, −60°, −90°, −120°, −225°, −240°, −270°, −3…Free
  2. Q2State the sign of $\tan 380°$.Free
  3. Q3State the sign of $\cot 230°$.Free
  4. Q4State the sign of $\sec 468°$.Preview
  5. Q5State the signs of $\cos 4^c$ and $\cos 4°$. Which of these two is greater?Preview
  6. Q6State the quadrant in which θ lies if $\sin\theta < 0$ and $\tan\theta > 0$.Preview
  7. Q7State the quadrant in which θ lies if $\cos\theta < 0$ and $\tan\theta > 0$.Preview
  8. Q8Evaluate: $\sin 30° + \cos 45° + \tan 180°$.Preview
  9. Q9Evaluate: $\text{cosec}\,45° + \cot 45° + \tan 0°$.Preview
  10. Q10Evaluate: $\sin 30° \times \cos 45° \times \tan 360°$.Preview
  11. Q11Find all trigonometric functions of the angle in standard position whose terminal arm passes through the point $(3, −4)$.Preview
  12. Q12If $\cos\theta = \dfrac{12}{13}$, $0 < \theta < \dfrac{\pi}{2}$, find the value of $\dfrac{\sin^2\theta − \cos^2\theta}{2\sin\theta\cos\thet…Preview
  13. Q13Using tables, evaluate: $4\cot 45° − \sec^2 60° + \sin 30°$.Preview
  14. Q14Using tables, evaluate: $\cos^2 0° + \cos^2\dfrac{\pi}{6} + \cos^2\dfrac{\pi}{3} + \cos^2\dfrac{\pi}{2}$.Preview
  15. Q15Find the other trigonometric functions if $\cos\theta = −\dfrac{3}{5}$ and $180° < \theta < 270°$.Preview
  16. Q16Find the other trigonometric functions if $\sec A = −\dfrac{25}{7}$ and $A$ lies in the second quadrant.Preview
  17. Q17Find the other trigonometric functions if $\cot x = \dfrac{3}{4}$, $x$ lies in the third quadrant.Preview
  18. Q18Find the other trigonometric functions if $\tan x = −\dfrac{5}{12}$, $x$ lies in the fourth quadrant.Preview
2.1.1

Trigonometric functions with the help of a circle (trigonometric ratios of any angle)

Recovering the right-triangle ratios geometrically. Take a right triangle with the right angle at the foot of a perpendicular, and an acute angle θ.

2.1.2

Signs of trigonometric functions in different quadrants

Since and for the point P(x, y) on the unit circle (and ), the sign of each trigonometric function in a given quadrant is decided entirely by the signs of the coordinates of P there — no separate rule…

2.1.3

Range of sinθ and cosθ

Let P(x, y) be a point on the unit circle, so , and let (with B the terminal ray through P). Since P lies on the unit circle,

2.1.4

Trigonometric functions of specific angles

The unit-circle definition lets us find exact trigonometric values at standard angles by locating the coordinates of the corresponding point P geometrically, rather than approximating.

2.1.5

Trigonometric functions of negative angles

Let P(x, y) be a point on the unit circle with . The angle (the same rotation but in the opposite direction) has its terminal ray meeting the unit circle at the mirror image of P across the x-axis, i.…

2.2

Fundamental Identities

A trigonometric identity is an equation that holds for every admissible value of θ — not just at a few special angles.

+Exercise 2.2i36 questions
  1. Q19If $2\sin A = 1 = \sqrt{2}\cos B$ and $\dfrac{\pi}{2} < A < \pi$, $\dfrac{3\pi}{2} < B < 2\pi$, then find the value of an expression in $\ta…Free
  2. Q20If $\sin A = \dfrac{3}{5}$ and $\sin B = \dfrac{4}{5}$ and $A, B$ are angles in the second quadrant, then prove that $4\cos A + 3\cos B = −5…Free
  3. Q21If $\tan\theta = \dfrac{1}{2}$, evaluate $\dfrac{2\sin\theta + 3\cos\theta}{4\cos\theta + 3\sin\theta}$.Free
  4. Q22Eliminate θ from: $x = 3\sec\theta$, $y = 4\tan\theta$.Preview
  5. Q23Eliminate θ from: $x = 6\,\text{cosec}\,\theta$, $y = 8\cot\theta$.Preview
  6. Q24Eliminate θ from: $x = 4\cos\theta − 5\sin\theta$, $y = 4\sin\theta + 5\cos\theta$.Preview
  7. Q25Eliminate θ from: $x = 5 + 6\,\text{cosec}\,\theta$, $y = 3 + 8\cot\theta$.Preview
  8. Q26Eliminate θ from: $2x = 3 − 4\tan\theta$, $3y = 5 + 3\sec\theta$.Preview
  9. Q27If $2\sin^2\theta + 3\sin\theta = 0$, find the permissible values of $\cos\theta$.Preview
  10. Q28If $2\cos^2\theta − 11\cos\theta + 5 = 0$, then find possible values of $\cos\theta$.Preview
  11. Q29Find the acute angle θ such that $2\cos^2\theta = 3\sin\theta$.Preview
  12. Q30Find the acute angle θ such that $5\tan^2\theta + 3 = 9\sec\theta$.Preview
  13. Q31Find $\sin\theta$ such that $3\cos\theta + 4\sin\theta = 4$.Preview
  14. Q32If $\text{cosec}\,\theta + \cot\theta = 5$, then evaluate $\sec\theta$.Preview
  15. Q33If $\cot\theta = \dfrac{4}{3}$ and $\pi < \theta < \dfrac{3\pi}{2}$ then find the value of $4\,\text{cosec}\,\theta + 5\cos\theta$.Preview
  16. Q34Find the Cartesian co-ordinates of the point whose polar co-ordinates are $(3, 90°)$.Preview
  17. Q35Find the Cartesian co-ordinates of the point whose polar co-ordinates are $(1, 180°)$.Preview
  18. Q36Find the polar co-ordinates of the point whose Cartesian co-ordinates are $(5, 5)$.Preview
  19. Q37Find the polar co-ordinates of the point whose Cartesian co-ordinates are $(1, \sqrt{3})$.Preview
  20. Q38Find the polar co-ordinates of the point whose Cartesian co-ordinates are $(−1, −1)$.Preview
  21. Q39Find the polar co-ordinates of the point whose Cartesian co-ordinates are $(−\sqrt{3}, 1)$.Preview
  22. Q40Find the value of $\sin\dfrac{19\pi}{3}$.Preview
  23. Q41Find the value of $\cos 1140°$.Preview
  24. Q42Find the value of $\cot\dfrac{25\pi}{3}$.Preview
  25. Q43Prove the following identity (the printed source scan is corrupted at the exact arrangement of this expression involving $\tan^2 A$ and $\si…Preview
  26. Q44Prove that $(\cos^2 A − 1)(\cot^2 A + 1) = −1$.Preview
  27. Q45Prove that $(\sin\theta + \sec\theta)^2 + (\cos\theta + \text{cosec}\,\theta)^2 = (1 + \text{cosec}\,\theta\,\sec\theta)^2$.Preview
  28. Q46Prove that $(1 + \cot\theta − \text{cosec}\,\theta)(1 + \tan\theta + \sec\theta) = 2$.Preview
  29. Q47Prove that $\dfrac{\tan^3\theta}{1+\tan^2\theta} + \dfrac{\cot^3\theta}{1+\cot^2\theta} = \sec\theta\,\text{cosec}\,\theta − 2\sin\theta\cos…Preview
  30. Q48Prove that $\dfrac{1}{\sec\theta − \tan\theta} − \dfrac{1}{\cos\theta} = \dfrac{1}{\cos\theta} − \dfrac{1}{\sec\theta + \tan\theta}$.Preview
  31. Q49Prove that $\dfrac{\sin\theta}{1+\cos\theta} + \dfrac{1+\cos\theta}{\sin\theta} = 2\,\text{cosec}\,\theta$.Preview
  32. Q50Prove that $\dfrac{\tan\theta}{\sec\theta − 1} = \dfrac{\sec\theta + 1}{\tan\theta}$.Preview
  33. Q51Prove that $\dfrac{\cot\theta}{\text{cosec}\,\theta − 1} = \dfrac{\text{cosec}\,\theta + 1}{\cot\theta}$.Preview
  34. Q52Prove that $(\sec A + \cos A)(\sec A − \cos A) = \tan^2 A + \sin^2 A$.Preview
  35. Q53Prove that $1 + 3\,\text{cosec}^2\theta\cdot\cot^2\theta + \cot^6\theta = \text{cosec}^6\theta$.Preview
  36. Q54Prove the following identity (the printed source scan is corrupted at the exact arrangement of this expression involving $\sec\theta$ and $\…Preview
2.2.1

Domain and Range of Trigonometric functions

Here sinθ, cosθ, tanθ (and their reciprocals) are treated as functions of a real variable θ, measured in radians.

2.2.2

Periodicity of Trigonometric functions

A function f is periodic if there is a constant such that for every x in its domain — so that . The smallest such positive p is called the fundamental period (or simply the period) of f.

2.2.3

Graphs of Trigonometric functions

All trigonometric functions are periodic, with sine and cosine repeating every radians and tangent repeating every radians. Plotting them over one period turns these algebraic facts into a picture.

2.2.4

Applications of the Fundamental Identities (worked examples)

These examples show the standard moves used with , , : squaring a given relation, forming a quadratic in one ratio, combining fractions over a common denominator, and eliminating θ between two paramet…

2.2.5

Polar Co-ordinate System

Cartesian co-ordinates locate a point by how far it is across and up from the origin, . Polar co-ordinates instead locate a point by its straight-line distance from a fixed point, and the direction of…

More questions

+Show 41 questions41 questions
  1. Q55The value of the expression $\cos 1° \cdot \cos 2° \cdot \cos 3° \cdots \cos 179°$ is: A) −1 B) 0 C) $\dfrac{1}{2}$ D) 1Free
  2. Q56An expression in $\tan A$ and $\sec A$ is equal to: A) $2\,\text{cosec}\,A$ B) $2\sec A$ C) $2\sin A$ D) $2\cos A$ (the printed source scan…Free
  3. Q57If α is a root of $25\cos^2\theta + 5\cos\theta − 12 = 0$, $\dfrac{\pi}{2} < \alpha < \pi$, then $\sin 2\alpha$ is equal to: A) $−\dfrac{24}…Free
  4. Q58If θ = 60°, then $\dfrac{1+\tan^2\theta}{2\tan\theta}$ is equal to: A) $\dfrac{2}{\sqrt3}$ B) $\dfrac{\sqrt3}{2}$ C) $\dfrac{1}{\sqrt3}$ D)…Preview
  5. Q59If $\sec\theta = m$ and $\tan\theta = n$, then $(m+n) + \dfrac{1}{m+n}$ is equal to: A) 2 B) $mn$ C) $2m$ D) $2n$Preview
  6. Q60If $\text{cosec}\,\theta + \cot\theta = \dfrac{5}{2}$, then the value of $\tan\theta$ is: A) $\dfrac{14}{25}$ B) $\dfrac{20}{21}$ C) $\dfrac…Preview
  7. Q61An expression in $\sin\theta$ and $\cos\theta$ equals: A) 0 B) 1 C) $\sin\theta$ D) $\cos\theta$ (the printed source scan is corrupted at th…Preview
  8. Q62If $\text{cosec}\,\theta − \cot\theta = q$, then the value of $\cot\theta$ is: A) $\dfrac{1+q^2}{2q}$ B) $\dfrac{1-q^2}{2q}$ C) $\dfrac{2q}{…Preview
  9. Q63The cotangents of the angles $\dfrac{\pi}{3}, \dfrac{\pi}{4}, \dfrac{\pi}{6}$ are in: A) A.P. B) G.P. C) H.P. D) Not in progressionPreview
  10. Q64The value of $\tan 1° \tan 2° \tan 3° \cdots \tan 89°$ is equal to: A) −1 B) 1 C) $\dfrac{\pi}{2}$ D) 2Preview
  11. Q65Find the trigonometric functions of $90°, 120°, 225°, 240°, 270°, 315°, −120°, −150°, −180°, −210°, −300°, −330°$.Preview
  12. Q66State the sign of $\text{cosec}\,520°$.Preview
  13. Q67State the sign of $\cot 1899°$.Preview
  14. Q68State the sign of $\sin 986°$.Preview
  15. Q69State the quadrant in which θ lies if $\tan\theta < 0$ and $\sec\theta > 0$.Preview
  16. Q70State the quadrant in which θ lies if $\sin\theta < 0$ and $\cos\theta < 0$.Preview
  17. Q71State the quadrant in which θ lies if $\sin\theta > 0$ and $\tan\theta < 0$.Preview
  18. Q72Which is greater, $\sin(1856°)$ or $\sin(2006°)$?Preview
  19. Q73Which of the following is positive? $\sin(−310°)$ or $\sin(310°)$Preview
  20. Q74Show that $1 − 2\sin\theta\cos\theta \geq 0$ for all $\theta \in \mathbb{R}$.Preview
  21. Q75Show that $\tan^2\theta + \cot^2\theta \geq 2$ for all $\theta \in \mathbb{R}$ for which both are defined.Preview
  22. Q76If $\sin\theta = \dfrac{x^2−y^2}{x^2+y^2}$ then find the values of $\cos\theta$, $\tan\theta$ in terms of $x$ and $y$.Preview
  23. Q77If $\sec\theta = \sqrt{2}$ and $\dfrac{3\pi}{2} < \theta < 2\pi$ then evaluate $\dfrac{1}{1+\tan\theta+\text{cosec}\,\theta} + \dfrac{1}{1+\…Preview
  24. Q78Prove that $\sin^2 A\cos^2 B + \cos^2 A\sin^2 B + \cos^2 A\cos^2 B + \sin^2 A\sin^2 B = 1$.Preview
  25. Q79Prove that $\dfrac{(1+\cot\theta+\tan\theta)(\sin\theta−\cos\theta)}{\sec^3\theta−\text{cosec}^3\theta} = \sin^2\theta\cos^2\theta$.Preview
  26. Q80Prove an identity involving $\tan\theta$, $\cos\theta$ and $\sin^2\theta$ (the printed source scan is corrupted at the exact arrangement of…Preview
  27. Q81Prove that $2\sec^2\theta − \sec^4\theta − 2\,\text{cosec}^2\theta + \text{cosec}^4\theta = \cot^4\theta − \tan^4\theta$.Preview
  28. Q82Prove that $\sin^4\theta + \cos^4\theta = 1 − 2\sin^2\theta\cos^2\theta$.Preview
  29. Q83Prove that $2(\sin^6\theta + \cos^6\theta) − 3(\sin^4\theta + \cos^4\theta) + 1 = 0$.Preview
  30. Q84Prove that $\cos^4\theta − \sin^4\theta + 1 = 2\cos^2\theta$.Preview
  31. Q85Prove that $\sin^4\theta + 2\sin^2\theta\cos^2\theta = 1 − \cos^4\theta$.Preview
  32. Q86Prove that $\dfrac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta} + \dfrac{\sin^3\theta−\cos^3\theta}{\sin\theta−\cos\theta} = 2$.Preview
  33. Q87Prove that $\tan^2\theta − \sin^2\theta = \sin^4\theta\sec^2\theta$.Preview
  34. Q88Prove that $(\sin\theta + \text{cosec}\,\theta)^2 + (\cos\theta + \sec\theta)^2 = \tan^2\theta + \cot^2\theta + 7$.Preview
  35. Q89Prove that $\sin^8\theta − \cos^8\theta = (\sin^2\theta − \cos^2\theta)(1 − 2\sin^2\theta\cos^2\theta)$.Preview
  36. Q90Prove that $\sin^6 A + \cos^6 A = 1 − 3\sin^2 A + 3\sin^4 A$.Preview
  37. Q91Prove that $(1 + \tan A\tan B)^2 + (\tan A − \tan B)^2 = \sec^2 A\sec^2 B$.Preview
  38. Q92Prove an identity involving $\cot\theta$ and $\text{cosec}\,\theta$ (the printed source scan is corrupted at the exact arrangement of this e…Preview
  39. Q93Prove that $\dfrac{\tan\theta+\sec\theta−1}{\tan\theta−\sec\theta+1} = \sec\theta+\tan\theta$.Preview
  40. Q94Prove an identity involving $\text{cosec}\,\theta$, $\cot\theta$, $\sin\theta$ and $\cos\theta$ (the printed source scan is corrupted at the…Preview
  41. Q95Prove an identity involving $\text{cosec}\,\theta$ and $\cot\theta$ (the printed source scan is corrupted at the exact arrangement of this e…Preview