# Trigonometric Functions:Class 11 Maths NCERT Chapter 3

## Key Features of NCERT Material for Class 11 Maths Chapter 3 – Trigonometric Functions

In the previous chapter 2:Relations and functions we have learned about the relations between the two objects in the pair.In the previous Chapter 3:Trigonometric functions we will study about the concept of trignometric ratios to trignometric functions.

### Quick revision notes

#### Angle

Angle is a measure of rotation of a given beam about its underlying point. The first beam is known as the underlying side and the last situation of beam after rotation is called terminal side of the angle. The purpose of rotation is called vertex. If the bearing of rotation is hostile to clockwise, the angle is supposed to be positive and if the heading of rotation is clockwise, at that point the angle is negative.

#### Estimating Angles

There are two frameworks of estimating angles

Sexagesimal framework (degree measure): If a rotation from the underlying side to terminal side is of a revolution, the angle is said to have a measure of one degree, composed as 1°.

One sixtieth of a degree is known as a moment, composed as 1′ and one-sixtieth of a moment is known as a second, composed as 1″

In this manner, 1° = 60′ and 1′ = 60″

Circular system(radian measure): A radian is an angle subtended at the focal point of a circle by a curve, whose length is equivalent to the sweep of the circle. We mean 1 radian by 1°.

#### Connection Between Radian and Degree

We realize that a total circle subtends at its middle an angle whose measure is 2π radians just as 360°.

or on the other hand 1 radian = 57° 16′ 21″ (approx)

#### Six Fundamental Trigonometric Identities

• sinx = • cos x = • tan x = • sin2 x + cos2 x = 1
• 1 + tan2x = sec2 x
• 1 + cot2 x = cosec2 x

Trigonometric Functions – Class 11 Maths Notes

Trigonometric ratios are characterized for intense angles as the proportion of the sides of a privilege angled triangle. The expansion of trigonometric proportions to any angle regarding radian measure (genuine number) are called trigonometric functions. The indications of trigonometric ratios in different quadrants have been given in the following table.

 I II III IV Sin x + + – – Cos x + – – + Tan x + – + – Cosec x + + – – Sec x + – – + Cot x + – + –

### Domain and Range of Trigonometric Functions

 Functions Domain Range Sine R [-1, 1] Cos R [-1, 1] Tan R – {(2n + 1)  $latex-php-14$ : n ∈ Z R Cot R – {nπ: n ∈ Z} R Sec R – {(2n + 1) : n ∈ Z R – (-1, 1) Cosec R – {nπ: n ∈ Z} R – (-1, 1)

Sine, Cosine, and Tangent of Some Angles Less Than 90° #### Allied or Related Angles

The angles are called allied or related angle and θ ± n × (2π) are called coterminal angles. For general decrease, we have adhering to rules, the estimation of trigonometric function for ( ) is mathematically equivalent to

• the estimation of a similar function, if n is an even number with the mathematical indication of the function according to the quadrant where angle lies.
• the relating co-function of θ, if n is an odd whole number with the mathematical indication of the function for the quadrant where it lies, here sine and cosine, tan and bed, sec and cosec are cofunctions of one another

#### Functions of Negative Angles

For any intense angle of θ. We have,

• sin(- θ) = – sinθ
• cos (- θ) = cosθ
• tan (- θ) = – tanθ
• cot (- θ) = – cotθ
• sec (- θ) = secθ
• cosec (- θ) = – cosecθ

#### A few Formulae Regarding Compound Angles

An angle comprised of the whole or difference of at least two angles is called compound angles. The fundamental outcomes in bearing are called trigonometric identities as given underneath:

1. sin (x + y) = sin x cos y + cos x sin y
2. sin (x – y) = sin x cos y – cos x sin y
3. cos (x + y) = cos x cos y – sin x sin y
4.  cos (x – y) = cos x cos y + sin x sin y (ix) sin(x + y) sin (x – y) = sin2 x – sin2 y = cos2 y – cos2 x

(x) cos (x + y) cos (x – y) = cos2 x – sin2 y = cos2 y – sin2 x

#### Transformation Formulae

• 2 sin x cos y = sin (x + y) + sin (x – y)
• 2 cos x sin y = sin (x + y) – sin (x – y)
• 2 cos x cos y = cos (x + y) + cos (x – y)
• 2 sin x sin y = cos (x – y) – cos (x + y)
• sin x + sin y = 2 sin( ) cos( )
• sin x – sin y = 2 cos( ) sin( )
• cos x + cos y = 2 cos( ) cos( )
• cos x – cos y = -2 sin( ) sin( )

#### Trigonometric Ratios of Multiple Angles ###  #### Product of Trigonometric Ratios

• sin x sin (60° – x) sin (60° + x) = sin 3x
• cos x cos (60° – x) cos (60° + x) = cos 3x
• tan x tan (60° – x) tan (60° + x) = tan 3x
• cos 36° cos 72° = • cos x . cos 2x . cos 22x . cos 23x … cos 2n-1 = #### Sum of Trigonometric Ratio, if Angles are in A.P. #### Trigonometric Equations

Equation which includes trigonometric functions of obscure angles is known as the trigonometric equation.

#### Arrangement of a Trigonometric Equation

An answer of a trigonometric equation is the estimation of the obscure angle that fulfills the equation.

A trigonometric equation may have an unbounded number of arrangements.

Principal Solution

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

General Solutions

An answer of a trigonometric equation, including ‘n’ which gives all arrangement of a trigonometric equation is known as the general arrangements.

General Solutions of Trigonometric Equation

• sin x = 0 ⇔ x = nπ, n ∈ Z
• cos x = 0 ⇔ x = (2n + 1) , n ∈ Z
• tan x = 0 ⇔ x = nπ, n ∈ Z
• sin x = sin y ⇔ x = nπ + (-1)n y, n ∈ Z
• cos x = cos y ⇔ x = 2nπ ± y, n ∈ Z
• tan x = tan y ⇔ x = nπ ± y, n ∈ Z
• sin2 x = sin2 y ⇔ x = nπ ± y, n ∈ Z
• cos2 x = cos2 y ⇔ x = nπ ± y, n ∈ Z
• tan2 x = tan2 y ⇔ x = nπ ± y, n ∈ Z

#### Basic Rules of Triangle In a triangle ABC, the angles are indicated by capital letters A, B and C and the lengths of sides of inverse to these angles are meant by little letters a, b and c, separately.

Sine Rule Cosine Rule

a2 = b2 + c2 – 2bc cos A

b2 = c2 + a2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

Projection Rule

a = b cos C + c cos B

b = c cos A + a cos C

c = a cos B + b cos A

October 30, 2020