Chapter 3

Trigonometric Functions

The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron', and it means ‘measuring the sides of a triangle’. The subject was originally developed to solve geometric problems involving triangles. Trigonometry is an essential section of mathematics; trigonometric functions and identities are used in solving most mathematical problems. If students don't have a clear understanding of this chapter, they will face trouble in solving questions of other chapters too. In class 10, basic knowledge of trigonometric identities and applications is already taught to students. In Class 11 trigonometry, students will learn about trigonometric ratios and functions and their properties. From an examination point of view also, this chapter is important.

Topics covered in the chapter: Trigonometric functions 

     1. Introduction 

     2. sequences

     3. series

     4. Arithmetic progression(A.P)

     5. Arithmetic mean

     6. Geometric progression (G.P)

     7. General term of a G.P

     8. Sum of n terms of a G.P

     9. Geometric mean (G.M)

    10. Relationship between A.M and G.M

     11. Sum of n terms of special series

 

Introduction

In this chapter, students will learn about various trigonometric functions and how to use them in solving mathematical problems. We will also be discussing sequences, series, arithmetic progression, and geometric progression, and other related topics.

Formulas of trigonometry class 11

1. cos2 x + sin2 x = 1

2. 1 + tan2 x = sec2 x 

3. 1 + cot2 x = cosec2 x

4. cos (2nπ + x) = cos x 

5. sin (2nπ + x) = sin x

6. sin (– x) = – sin x 

7. cos (– x) = cos x

8. cos (x + y) = cos x cos y – sin x sin y

9. cos (x – y) = cos x cos y + sin x sin y

10.  cos ( π 2 − x ) = sin x

11.  sin ( π 2 − x ) = cos x

12.  sin (x + y) = sin x cos y + cos x sin y 

13.  sin (x – y) = sin x cos y – cos x sin y 

14. cos (π/2+ x )  = – sin x

15. cos (π – x) = – cos x

16. cos (π + x) = – cos x

17. cos (2π – x) = cos x

18. sin (π/ 2+ x )  = cos x

19. sin (π – x) = sin x

20. sin (π + x) = – sin x

21. sin (2π – x) = – sin x

22. If none of the angles x, y and (x ± y) is an odd multiple of π 2, then 

                             tan(x+y) = tan x + tan y / 1-tan x tan y

                             tan(x-y) = tan x + tan y / 1+ tan x tan y

23. If none of the angles x, y and (x ± y) is a multiple of π, then

                cot (x + y) = cot x cot y -  1 / cot x + cot y 

                cot (x – y) = cot x cot y - 1/ cot x - cot y 

24. cos 2x = cos2x – sin2 x = 2cos2x – 1 = 1 – 2 sin2x = 1 - tan2 x / 1 +tan2 x 

25. sin 2x = 2 sin x cos x= 2 tan x/1+tan2x

26. sin 3x = 3sinx – 4sin3x

27. cos 3x = 4cos3x – 3cos x

28. 2cos x cos y = cos ( x + y) + cos ( x – y)

29. – 2sin x sin y = cos (x + y) – cos (x – y)

30. 2sin x cos y = sin (x + y) + sin (x – y)

31. 2 cos x sin y = sin (x + y) – sin (x – y)

32. tan 3x = 3 tanx- tan3x/1-3 tan2x  

33. cos x + cos y = 2cos( x+y/2 )cos (x-y/2)

34. cos x – cos y = – 2sin(x+y/2) sin (x-y/2 )

 

In general, a sequence is an ordered list of numbers or items. 

In mathematical terms, 'sequence' is defined as a function that has a domain belonging to a set of natural numbers. 

The numbers or items of sequence are called 'elements' or 'terms'.

When we add all the elements or terms of the sequence, we get a series. The value we obtain by adding all the elements or terms of a sequence is called the value of the series.  For example, the sequence "2, 3, 4,5" contains the terms "2", "3", "4", and "5"; the corresponding series is the sum " 2 + 3 + 4+5", and the series' value is 14.

An arithmetic progression is a sequence of numbers. In A.P, each successive term is equal to the sum of all of its preceding terms and a fixed number, called the common difference(d). 

The progression of A.P is decided by the fixed number. If the fixed number is a positive value, then A.P will increase; if it’s a negative integer, A.P will decrease. 

The first term of A.P is represented by  ‘a’.

The Nth term of an AP:

    • tn= a+(n−1)d

Where, d = an−an−1

Sum of first N terms of an AP:

Sn= n/2 [a+(n−1)d]= n/2[a+l]

Where,l is last term of an A.P

G.P is also a type of sequence. In G.P, every term is calculated by multiplying its preceding term by a fixed constant number called the common ratio. 

First-term of GP is always a nonzero integer.

Common ratio = term/its preceding term  

a = first term 

r= common ratio 

Then, 

G.P = a,ar,ar2,ar3,ar4,.......

The Nth term of a GP:

tn =arn−1

Sum of first N terms of a GP: Sn=a(1−rn)/(1−r),r≠1

Sum of infinite GP when |r|<1&n→∞

|r|<1⇒rn→0⇒S°°=a/1−r

For two positive numbers, let say m and n, the arithmetic mean is always equal to or greater than the geometric mean, and geometric mean is equal to  √mn.

AM or Arithmetic Mean is the average of the sets of numbers of the data. It is calculated by adding all the elements of the data divided by the total number of elements in the data.

GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression.

 

There are a total of three special series. 

    1. 1 + 2 + 3 +… + n (sum of first n natural numbers)

    2. 12 + 22 + 32 +… + n2(sum of squares of the first n natural numbers)

    3. 13 + 23 + 33 +… + n3(sum of cubes of the first n natural numbers)

Sum to n terms of first n natural numbers: 

 

Sum of First n Natural Numbers (1 + 2 + 3 +… + n)

This is an A.P. 

Here,  a = 1, d =1 

Formula of the special series

Sn = n/2 [2a + ( n-1 )d]

Sn = n/2 [2× 1 + ( n-1 )1]

= n/2 [ 2+n-1]

= n/2 [n+1]

 

Here are some trigonometry Ncert solutions class 11

Q1. tan x = √3

Solution:

NCERT Solutions for Class 11 Chapter 3 Ex 3.4 Image 1

Q2 . sin 2x + cos x = 0

Solution:

It is given that

sin 2x + cos x = 0

We can write it as

2 sin x cos x + cos x = 0

cos x (2 sin x + 1) = 0

cos x = 0 or 2 sin x + 1 = 0

Let cos x = 0

NCERT Solutions for Class 11 Chapter 3 Ex 3.4 Image 10

 

     1. What are the basic trigonometric functions?

 The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

     2. How many exercises are there in Chapter 3 of NCERT Solutions for Class 11 Maths?

Altogether, there are four exercises in chapter 3 of NCERT Class 11 maths, including one miscellaneous exercise.

     3. How are Radians and Degrees related?

Degree measures angle in 2 dimensions or upto 360°.

 Radians measure angles in 3 dimensions. It is represented by (theta). 

Radian = arc length (s) / radius(r). 

The relation between Radian and degree is given in terms of a circle.

360° =2pi radian(theta).

Other Courses

Related Chapters