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CBSE NCERT NOTES CLASS 11 MATHEMATICS CHAPTER 3

TRIGONOMETRIC FUNCTIONS

Chapter Notes

Measurement of Angle

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex.

If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative

Degree: If a rotation from the initial side to terminal side is $\frac{1}{360}$ ^th of a revolution, the angle is said to have a measure of one degree (1°).

One degree is divided into 60 minutes, and a minute is divided into 60 seconds. That is, one sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.

Radian: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

One complete revolution of the initial side subtends an angle of 2π radian.

In a circle of radius r, an arc of length l will subtend an angle θ radian at the centre, given by,

$θ = \frac{l}{r} rad or l = r θ .$

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$2π radian = 360 °$

$or π radian = 180 °$

$1 radian = \frac{180 °}{π} = 57 ° 16^{'} (approximately)$

$1 ° = \frac{π}{180} = 0.01746 radians approximately$

$Radian measure = \frac{π}{180} \times Degree measure$

$Degree measure \frac{180}{π} \times Radian measure$

1° = 60′ and 1′= 60′′

Some Important Angle Measures in Degree and Radian

$d e g$	$0$	$30$	$45$	$60$	$90$	$180$	$270$	$360$
$r a d$	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$π$	$\frac{3 π}{2}$	$2 π$

Relation between radian and real numbers

Radian measures and real numbers can be considered as one and the same. Consider the line PAQ which is tangent to the circle at A. Let the point A represent the real number zero, AP represents positive real numbers and AQ represents negative real numbers. If we rope the line AP in the anticlockwise direction along the circle, and AQ in the clockwise direction, then every real number will correspond to a radian measure and conversely.

Trigonometric Functions

Trigonometric ratios for an angle are the ratio of sides of a right angled triangle.

$\sin x = \frac{p}{h}$

$\cos x = \frac{b}{h}$

$c o s e c x = \frac{h}{p} = \frac{1}{\sin x} x \neq  n π$

$\tan x = \frac{p}{b} = \frac{\sin x}{\cos x} x \neq (2 n + 1) \frac{π}{2}$

$\sec x = \frac{h}{b} = \frac{1}{\cos x} x \neq (2 n + 1) \frac{π}{2}$

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$\cot x = \frac{b}{p} = \frac{\cos x}{\sin x}, x \neq n π$

$\sin x = 0 \Rightarrow x = n π$

$\cos x = 0 \Rightarrow x = (2 n + 1) \frac{π}{2}, where n is some integer$

sin² x + cos² x = 1

1 + tan² x = sec² x

1 + cot² x = cosec² x

Values of trigonometric ratios of some common angles

	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$π$	$\frac{3 π}{2}$	$2 π$
$s i n$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$0$	$- 1$	$0$
$c o s$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$- 1$	$0$	$1$
$t a n$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$n d$	$0$	$n d$	$0$
$c o t$	$n d$	$3$	$1$	$\frac{1}{\sqrt{3}}$	$0$	$n d$	$0$	$n d$
$s e c$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	$n d$	$- 1$	$n d$	$1$
$c o s e c$	$n d$	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$	$n d$	$- 1$	$n d$

Signs of Trigonometric Functions

Trends of Trigonometric Functions

Q I

Q II

Q III

Q IV

sin

increases from

0 to 1

decreases from

1 to 0

decreases from

0 to -1

increases from

-1 to 0

cos

decreases from

1 to 0

decreases from

0 to -1

increases from

-1 to 0

increases from

0 to 1

tan

increases from

0 to ∞

increases from

-∞ to 0

increases from

0 to ∞

increases from

-∞ to 0

cot

decreases from

∞ to 0

decreases from

0 to -∞

decreases from

∞ to 0

decreases from

0 to -∞

sec

increases from

1 to ∞

increases from -∞ to -1

decreases from -1 to -∞

decreases from

∞ to 1

cosec

decreases from

∞ to 1

increases from

1 to ∞

increases from

-∞ to -1

decreases from

-1 to-∞

Graphical Representation of Trigonometric Functions

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Important Trigonometric Formulae

$\sin (- x) = - \sin x$	$c o s e c (- x) = - c o s e c x$
$\cos (- x) = \cos x$	$\sec (- x) = \sec x$
$\tan (- x) = - \tan x$	$\cot (- x) = - \cot x$
$\cos (\frac{π}{2} - x) = \sin x$	$\sec (\frac{π}{2} - x) = c o s e c x$
$\sin (\frac{π}{2} - x) = \cos x$	$c o s e c (\frac{π}{2} - x) = \sec x$
$\tan (\frac{π}{2} - x) = \cot x$	$\cot (\frac{π}{2} - x) = \tan x$
$\cos (\frac{π}{2} + x) = - \sin x$	$\sec (\frac{π}{2} + x) = - c o s e c x$
$\sin (\frac{π}{2} + x) = \cos x$	$c o s e c (\frac{π}{2} + x) = \sec x$
$\tan (\frac{π}{2} + x) = - \cot x$	$\cot (\frac{π}{2} + x) = - \tan x$
$\cos (π - x) = - \cos x$	$\sec (π - x) = - \sec x$
$\sin (π - x) = \sin x$	$c o s e c (π - x) = c o s e c x$
$\tan (π - x) = - \tan x$	$\cot (π - x) = - \cot x$
$\cos (π + x) = - \cos x$	$\sec (π + x) = - \sec x$
$\sin (π + x) = - \sin x$	$c o s e c (π + x) = - c o s e c x$
$\tan (π + x) = \tan x$	$\cot (π + x) = \cot x$
$\cos (\frac{3 π}{2} + x) = \sin x$	$\sec (\frac{3 π}{2} + x) = c o s e c x$
$\sin (\frac{3 π}{2} + x) = - \cos x$	$c o s e c (\frac{3 π}{2} + x) = - \sec x$
$\tan (\frac{3 π}{2} + x) = - \cot x$	$\cot (\frac{3 π}{2} + x) = - \tan x$
$\cos (2 π - x) = \cos x$	$\sec (2 π - x) = \sec x$
$\sin (2 π - x) = - \sin x$	$c o s e c (2 π - x) = - c o s e c x$
$t a n (2 π + x) = - t a n x$	$c o t (2 π + x) = - c o t x$

$\cos (x + y) = \cos x \cos y - \sin x \sin y$
$\cos (x - y) = \cos x \cos y + \sin x \sin y$
$\sin (x + y) = \sin x \cos y + \cos x \sin y$
$\sin (x - y) = \sin x \cos y - \cos x \sin y$
If none of the angles x, y and (x + y) is an odd multiple of $\frac{π}{2}$ , then $t a n (x + y) = \frac{t a n x + t a n y}{1 - t a n x t a n y}$ $t a n (x - y) = \frac{t a n x - t a n y}{1 + t a n x t a n y}$
If none of the angles x, y and (x + y) is a multiple of π, then $c o t (x + y) = \frac{c o t x c o t y - 1}{c o t y + c o t x}$ $c o t (x - y) = \frac{c o t x c o t y + 1}{c o t y - c o t x}$
cos 2x = cos²x – sin² x = 2 cos² x – 1 = 1 – 2 sin² x $= \frac{1 - {t a n}^{2} x}{1 + {t a n}^{2} x}, x n π + \frac{π}{2}$
sin 2x = 2 sin x cos x $= \frac{2 t a n x}{1 + {t a n}^{2} x}, x n π + \frac{π}{2}$
$t a n 2 x = \frac{2 t a n x}{1 - {t a n}^{2} x}, 2 x n π + \frac{π}{2}$
sin 3x = 3 sin x – 4 sin³x
cos 3x = 4 cos³ x – 3 cos x $t a n 3 x = \frac{3 t a n x - {t a n}^{3} x}{1 - 3 {t a n}^{2} x}$ $c o s x + c o s y = 2 c o s \frac{x + y}{2} c o s \frac{x - y}{2}$ $c o s x - c o s y = - 2 s i n \frac{x + y}{2} s i n \frac{x - y}{2}$ $s i n x + s i n y = 2 s i n \frac{x + y}{2} c o s \frac{x - y}{2}$ $s i n x - s i n y = 2 c o s \frac{x + y}{2} s i n \frac{x - y}{2}$ 2 cos x cos y = cos (x + y) + cos (x – y) –2 sin x sin y = cos (x + y) – cos (x – y) 2 sin x cos y = sin (x + y) + sin (x – y) 2 cos x sin y = sin (x + y) – sin (x – y)

Trigonometric Equations

Equations involving trigonometric functions of a variable are called trigonometric equations.

The solutions of a trigonometric equation for which 0 ≤ x < 2π are called principal solutions.

The expression involving integer ‘n’ which gives all solutions of a trigonometric equation is called the general solution.

Theorem 1 - For any real numbers x and y,

sin x = sin y ⇒ x = nπ + (–1)ⁿ y, where n ∈ Z

Theorem 2 - For any real numbers x and y,

cos x = cos y, ⇒ x = 2nπ ± y, where n ∈ Z

Theorem 3 - If x and y are not odd mulitple of $\frac{π}{2}$ , then

tan x = tan y ⇒ x = nπ + y, where n ∈ Z

Formulae for angles of a triangle

Sine formula

In any triangle, sides are proportional to the sines of the opposite angles. Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Cosine formulae

Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

a² = b²+ c²– 2bc cos A

b²= c²+ a² – 2ca cos B

c²= a² + b²– 2ab cos C

That is,

$\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$

$\cos B = \frac{c^{2} + a - b^{2}}{2 a c}$

$\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$

Tips for use of sine and cosine formulae

$(i) \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = k$

$(i i) \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k$

(iii) A + B + C = π

$(i v) \frac{A + B + C}{2} = \frac{π}{2}$

(v) Use formulae for π - x and $\frac{π}{2}$ $- x$

Some Other Useful Concepts

Componendo

If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a + b}{b} = \frac{c + d}{d}$

Dividendo

If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a - b}{b} = \frac{c - d}{d}$

Componendo and Dividendo

If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ .

Invertendo

If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{b}{a} = \frac{d}{c}$

Alternetendo

If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a}{c} = \frac{b}{d}$