Trigonometric Functions Class 11

Trigonometric functions for class 11 notes are provided here. All the concepts in class 11 trigonometric functions are covered here as per the CBSE syllabus, which helps students while preparing for the examinations. Go through the topics given below and score good marks in the final examination. In this article, we have discussed the topics like six important trigonometric functions, angle measures such as radians and degrees, sum and difference of two angles with many solved examples.

Trigonometric Functions Class 11 Topics

The topics and subtopics covered in trigonometric functions class 11 include:

  • Introduction
  • Angles
  • Trigonometric Functions
  • Trigonometric Functions of Sum and Difference of Two Angles

Introduction to Trigonometry

The word “ Trigonometry” is derived from the Greek word “trigon” and “metron”, which means measuring the sides of a triangle. Trigonometry is a branch of Mathematics, which helps to study the relationship between the angles and side lengths of a right triangle. In the earlier classes, we have learnt the different trigonometric ratios of the right triangle. In trigonometric functions class 11, we are going to study the generalised concepts of the trigonometric functions and its properties.

Angles

An angle is a measure, which gives the rotation of a ray about its initial position. In other words, the measurement of an angle is the amount of rotation performed to obtain the terminal side from the initial side. The angle can be measured in two different ways, such as 

Degree: If rotation of the ray from initial side to the terminal side is (1/360)th of the revolution, then the angle is considered to have a measure of one degree (1°).

Radians: The angle subtended at the centre by an arc length of one unit in a unit circle is considered to have a measure of 1 radian.

Relation Between Degree and Radian

As we know that the circle subtends at the centre of an angle whose radian measures 2π, and its degree measures 360 degrees. 

Hence, we get

2π = 360 degrees 

Or 

π radian = 180 degrees.

Therefore, 1 Radian = 180°/π 

Hence, 1 Radian = 57°16’ approximately

Also, 1 degree can be written as 

1° = π/180 radian = 0.01746 radians approximately.

Trigonometric Functions

The trigonometric functions are also called the angle functions, which relates the angles and the ratios of the sides of a right angle triangle. Considering the unit circle, the six important trigonometric functions are given as follows:

If “n” is an integer, then

Sin x =0 ⇒ x =nπ

Cos x = 0 ⇒ x= (2n+1)π/2

The other trigonometric functions in terms of sine and cosine functions are given as follows:

 Tan x = sin x /cos x ⇒ x≠ (2n+1)π/2

Cosec x = 1/sin x x ≠ nπ,

Sec x = 1/cos x ⇒  x≠ (2n+1)π/2

Cot x = cos x/sin x x ≠ nπ

Sum and Difference of Two Angles

Some of the important expressions for the trigonometric functions of the sum and difference of two angles and related expressions are given as follows:

  • Sin (-x) = – sin x
  • Cos (-x) = cos x
  • Cos (x+y) = cos x cos y – sin x sin y
  • Cos (x-y) = cos x cos y + sin x sin y
  • Cos (π/2 -x )= sin x
  • Sin (π/2 -x ) = cos x
  • sin (x+y) = sin x cos y + cos x sin y
  • sin (x-y) = sin x cos y – cos x sin y
  • Cos (π/2 + x ) = -sin x
  • Sin (π/2 + x ) = cos x
  • Cos (π-x) = -cos x 
  • Sin (π-x) = sin x
  • Cos (π+x) = -cos x
  • Sin (π+x) = -sin x
  • Cos (2π-x) = cos x
  • Sin (2π-x) = -sin x
  • Tan (x+y)= tan x + tan y /(1-tan x tan y)
  • Tan (x-y)= tan x – tan y /(1+tan x tan y)
  • Cot (x+y) = (cot x cot y -1)/(cot x + cot y)
  • Cot (x-y) = (cot x cot y +1)/(cot y – cot x)
  • Cos 2x = (1-tan2x)/(1+tan2x)
  • Sin 2x = 2tanx/(1+tan2x)
  • Tan 2x = 2tanx/(1-tan2x) 
  • Sin 3x = 3 sin x – 4 sin3x
  • Cos 3x =  4 cos3x – 3 cos x
  • Tan 3x = (3 tanx -tan3x)/(1-3tan2x)

Trigonometric Functions Class 11 Examples

Go through the below given trigonometric functions class 11 solved problems:

Example 1:

Convert  6 radians into the degree measure.

Solution:

Given : Radian = 6

We know that, π radian = 180°

Therefore 1 radian = 180°/π

Hence, 6 radians = (180°/π) × 6 degrees.

= (1080×7)/22 degrees

As we know, 1° = 60’

By simplifying the above equation, we get

 =343° 38′ 11″ approximately.

Thus, 6 radians is approximately equal to 343° 38′ 11″.

Example 2: 

Calculate the values of the other five trigonometric functions, if cot x = -5/12, where x lies in the second quadrant.

Solution:

Given: cot x = -5/12

From the above ratio, we can write, tan x = -12/5

Sec2x = 1 + tan2x

= 1+ (12/5)2

= 1+ (144/25)

=196/25

Therefore, sec x = = ±13/5

SInce, x lies in the second quadrant, the value of secant will be negative.

Hece, sec x = -13/5

Thus, cos x = -5/13

We know that, tan x = sin x/cos x

Sin x = tan x cos x

Sin x = (-12/5) × (-5/13)

Sin x = 12/13

Thus, cosec x = 13/12

Therefore, the value of trigonometric functions are:

Sin x = 12/13, cos x = -5/13, tanx = -12/5, cosex x = 13/12, sec x = -13/5

Example 3: 

Calculate the value of cos (–1710°).

Solution:

Given: cos (–1710°)

We know that the value of cos x repeats after the interval range of 2π, we can write:

cos (–1710°) =  cos [–1710° + (5 × 360°)]

cos (–1710°) = cos [–1710° + 1800°]

cos (–1710°) = cos 90° = 0

Therefore, the value of cos (–1710°) is 0.

Example 4: 

Prove that tan 3x tan 2x tan x = tan 3x – tan 2 x – tan x.

Solution:

To prove: tan 3x tan 2x tan x = tan 3x – tan 2 x – tan x

We know that 3x can be written as 2x+x

Hence, tan 3x = tan (2x+x)

By using the trigonometric identity, the above expression is written as:

Tan 3x = (tan 2x + tanx)/(1-tan 2x tanx)

Now, cross multiply above expression, we get

Tan 3x – tan 3x tan 2x tan x = tan 2x + tan x

Simplify the above equation, we get

tan 3x tan 2x tan x = tan 3x – tan 2x – tan x.

Hence, tan 3x tan 2x tan x = tan 3x – tan 2x – tan x is proved.

Practice Problems

Solve the following class 11 trigonometric functions problems given below:

  1. Find the length of the minor arc of the chord, whose circle diameter is 40 cm and chord length is 20 cm.
  2. Determine the degree measure for the corresponding radian measure: (a) -4   (b)7π/6
  1. Find the value of the other trigonometric functions, if cos x = -⅗, where x lies in the third quadrant.
  2. Find the value of sin 31π/3.
  3. Prove that [sin (x+y)] /[sin (x-y)] = (tan x+tan y)/(tan x- tan y)

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