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Class 11 Maths Revision Notes Trigonometric Functions

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Course

Mathematics – Iii

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Revision Notes

Class 11 Maths

Chapter 3 – Trigonometric Functions

TRIGONOMETRIC RATIOS & IDENTITIES

1. The meaning of Trigonometry

Tri Gon Metron

  

3 sides Measure

As a result, this area of mathematics was established in the ancient past to measure

a triangle's three sides, three angles, and six components. Time-trigonometric

functions are utilised in a variety of ways nowadays. The sine and cosine of an

angle in a right-angled triangle are the two fundamental functions, and there are

four more derivative functions.

2. Basic Trigonometric Identities

(a) sin 2   cos 2  1:  1 sin  1;  1 cos  1   R

(b) sec   tan  1:| sec | 1  R

2 2

(c) cosec   cot  1:| cosec | 1  R

2 2

Trigonometric Ratios of Standard Angles

Angles(In

Degrees)

0  30  45  60  90  180  270  360 

Angles(In

radians)

0 

6

4

3

2

 

2
32 

Sin 0

2
1
2
1
2

3 1 0 -1 0

Cos 1

2
3
2
1
2

1 0 -1 0 1

Tan 0

3

1 1 3 Not

Defined

0 Not

Defined

1

Cot Not

Defined

3 1

3

1 0 Not

Defined

0 Not

Defined

Csc Not

Defined

2

3

2 1 Not

Defined

-1 Not

Defined

Sec 1

3

2 2 2 Not

Defined

-1 Not

Defined

1

The relation between these trigonometric identities with the sides of the triangles

can be given as follows:

 Sine (theta) = Opposite/Hypotenuse

 Cos (theta) = Adjacent/Hypotenuse

 Tan (theta) = Opposite/Adjacent

   
       
   

   

 

and
2
sin
2
sin cos cos
sin(    )  sin  and cos(  )  cos
   
        
   

   

 

and
2 2
tan cot cot tan

tan(   )   tan and cot(   ) c to

   
       
   

   

 

and
2 2
sec cosec cosec sec

sec(   )   sec and cosec(  ) cosec

sin(   )   sin and cos(   ) c so

   
        
   

   

 

and
2 2
sin cos co s sin
3 3

tan(    )  tan  and co (t   ) cot

   
      
   

   

 

and
2 2
tan cot cot tan
3 3

sec(   )   sec and cosec(  )  cosec

   
        
   

   

 

and
2
c
2
sec cosec cosec se
3 3
   
       
   

#######    

#######  

and s
2 2
sin cos cos in
3 3

sin(2   )   sin and c s(2o    ) cos

   
        
   

   

 

and
2 2
tan cot co t tan
3 3

tan(2   )   tan  and cot( 2   ) cot

   
       
   

   

 

and
2
sec
2
sec cosec cosec
3 3

sec(2    )  sec and cosec(2  )  cose c

sin(2    )  sin  and cos 2(    ) cos
tan(2    )  tan  and cot 2(   ) cot

sec(2    )  sec and cosec(2  ) cosec

4. Trigonometric Functions of Sum or Difference of Two Angles

(a) sin(A  B)  sin Acos B cos Asin B

(b) sin(A  B)  sin Acos B cos Asin B

(c) cos( A  B )  cos A cos B sin A sinB

(d) cos(A  B)  cos Acos B sin Asin B

(e)

 
A B
A B
A B
1 tan tan
tan( )
tan tan

(f)

 
1 tan A tan B
tan(A B)
tan A tan B

(g)

 
B A
A B
A B
cot cot
cot( )
cot cot 1

(f)

 
cot B cot A
cot(A B)
cot A cot B 1

(h) sin 2 A  sin 2 B  cos 2 B  cos 2 A  sin(A  B) sin(A B)

(i) cos 2 A  sin 2 B  cos 2 B  sin 2 A  cos(A  B) cos(A B)

(j)

  
  
  
A B B C C A
A B C
A B C A B C
1 tan tan tan tan tan tan
tan( )
tan tan tan tan tan tan

5. Multiple Angles and Half Angles

(b)  

 
2 2
sin C sin D 2cos sin
C D C D

(c)  

 
C D
C D C D
2 2
cos cos 2cos cos

(d)   

 
2 2
cos C cos D 2sin sin
C D C D

8. Important Trigonometric Ratios

(a) sin n   0;cos n  ( 1) ; tan n n   0 where n Z

(b) sin15 or  

  

12 2
sin cos 7 5
3 1

or

####### 

12
cos 5

;

cos15  or     

12 2
cos sin 7 5
3 1

or

####### 

12
sin 5
   
  
3 1
tan15 2 3 cot 75
3 1
   
  
3 1
tan 75 2 3 cot
3 1

(c)

10

sin or 

 
4
sin
5 1

& cos36 or 

 
5 4
cos
5 1

9. Conditional Identities

If A  B  C   then :

(i) sin 2 A  sin 2 B  sin 2C 4sin Asin Bsin C

(ii)   

2 2 2
sin A sin B sin C 4cos cos cos
A B C

(iii) cos 2 A  cos 2 B  cos 2C    1 4cos Acos BcosC

(iv)    

2 2 2
cos A cos B cos C 1 4sin sin sin
A B C

(v) tan A  tan B  tan C tan A tan Btan C

(vi)   

2 2 2 2 2 2
tan tan tan tan tan tan 1
A B B C C A

(vii)     

2 2 2 2 2 2
cot cot cot cot cot cot
A B C A B C

(viii) cot A cot B  cot B cot C  cot C cot A 1

10. Range of Trigonometric Expression

E  a sin  bcos

 
     
 

  

a
E a b sin( ), where ta n
2 2 b
 
     
 

  

b
E a b cos( ), where ta n
2 2 a

Hence for any real value of  ,  a  b E a b

2 2 2 2

The trigonometric functions are very important for studying triangles, light, sound

or wave. The values of these trigonometric functions in different domains and

ranges can be used from the following table:

Trigonometric Functions Domain Range

Sin x R   1 sin x 1

Cos x R   1 cos x 1

Tan x R  {(2n  1)  / 2, n I R

b. y cosx

x  R y;  [ 1,1]

(c) y tanx

 
      

  

x R n n Z y R
2
(2 1) ; ;

(d) y cotx

x  R  {n  ; n  z };y R

(e) y cosecx

x  R  {n  ; n  Z }; y  ( , 1]  [1, )

e.,    

####### 

2 4
sin
1

or   

   

4 4 4 4
, , , ,
3 9 11

As a result, the trigonometric equation can have an unlimited number of solutions

and is categorised as follows:

Principal solution

As we know, the values of sin x and cos xwill get repeated after an interval of 2 .

In the same way, the values of tan xwill get repeated after an interval of .

So, if the equation has a variable 0  x  2  , then the solutions will be termed as

principal solutions.

Example:

Find the principal solutions of the equation x 

2
sin
3

.

Solution: We know that, 

####### 

3 2
sin
3

Also,

 
   
 

 

3 3
sin sin
2

Now, we know that sin(   x ) sin x.

Hence,  

 

3 3 2
sin sin
2 3

Therefore, the principal solutions of x 

2
sin
3

are 

####### 

3

x and

3
2

.

General solution

A general solution is one that involves the integer 'n' and yields all trigonometric

equation solutions. Also, the character ' Z ' is used to denote the set of integers.

Find the solution of x  

2
sin
3

.

Solution: We know that 

3 2
sin
3

. Therefore,    

x
2 3
sin sin
3

Using the unit circle properties, we get

 
     
 

  

x
3 3 3
sin sin sin sin 4

Hence,

x
3
sin sin 4

Since, we know that for any real numbers x and y ,sin x sin y implies

x  n    ( 1) ny, where n Z.

So, we get,

 
    
 

x n
3
( 1)
n 4

14 Results

1. sin   0   n

2.     

2
cos 0 (2n 1)

3. tan   0   n

4. sin   sin     n  ( 1 )n , where

 
  
 

 

2
,
2

5. cos   cos     2n , where  [0, ]

6. tan   tan     n , where

 
   
 

 

2
,
2

7. sin 2   sin 2     n .

8. cos   cos   

2 2

n  .

9. tan   tan     n 

2 2

.

10.      

sin 1 (4n 1)
2

11. cos   1    2 n

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Class 11 Maths Revision Notes Trigonometric Functions

Course: Mathematics – Iii

39 Documents
Students shared 39 documents in this course

University: MATS University

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Class XI Maths 1
Revision Notes
Class 11 Maths
Chapter 3 – Trigonometric Functions
TRIGONOMETRIC RATIOS & IDENTITIES
1. The meaning of Trigonometry
Tri Gon Metron

3 sides Measure
As a result, this area of mathematics was established in the ancient past to measure
a triangle's three sides, three angles, and six components. Time-trigonometric
functions are utilised in a variety of ways nowadays. The sine and cosine of an
angle in a right-angled triangle are the two fundamental functions, and there are
four more derivative functions.
2. Basic Trigonometric Identities
(a)
sin cos 1: 1 sin 1; 1 cos 1 R
22
(b)
sec tan 1:| sec | 1 R
22
(c)
cosec cot 1:| cosec | 1 R
22
Trigonometric Ratios of Standard Angles
Angles(In
Degrees)
0
30
45
60
90
180
270
360

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