06 - Let's recap trigonometry - Class 10 - Maths Class 10 Notes | EduRev

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Class 10 : 06 - Let's recap trigonometry - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Trigonometry: Trigonometry is a branch of mathematics that studies the 
relationships between the sides and angles of triangles. 
 
Trigonometric Ratios: The ratios of the sides of a right triangle are called 
trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are  the three 
common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) 
are the reciprocal of the ratios sin, cos and tan respectively. These are defined for 
acute angle A in right angled triangle ABC below: 
 
 
 
Sine of ?A =  
 
 
 
Cosine of ?A =  
 
 
 
Tangent of ?A=  
 
 
 
Cosecant of ?A= 
 
 
 
Secant of ?A = 
 
 
Cotangent of ?A =  
Side adjacent to angle A 
 
Side adjacent to angle A 
 
C 
B 
A 
Side 
opposite 
to angle 
A 
Side 
adjacent 
to angle 
A 
Hypotenuse 
Hypotenuse  
Side opposite to angle A BC 
AC 
= 
Hypotenuse  
Side adjacent to angle A AB 
AC 
= 
Side opposite to angle A BC 
AB 
= 
Hypotenuse  
Side opposite to angle A 
AC 
BC 
= 
Hypotenuse  
Side adjacent to angle A 
AC 
AB 
= 
Side opposite to angle A 
AB 
BC 
= 
Page 2


 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Trigonometry: Trigonometry is a branch of mathematics that studies the 
relationships between the sides and angles of triangles. 
 
Trigonometric Ratios: The ratios of the sides of a right triangle are called 
trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are  the three 
common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) 
are the reciprocal of the ratios sin, cos and tan respectively. These are defined for 
acute angle A in right angled triangle ABC below: 
 
 
 
Sine of ?A =  
 
 
 
Cosine of ?A =  
 
 
 
Tangent of ?A=  
 
 
 
Cosecant of ?A= 
 
 
 
Secant of ?A = 
 
 
Cotangent of ?A =  
Side adjacent to angle A 
 
Side adjacent to angle A 
 
C 
B 
A 
Side 
opposite 
to angle 
A 
Side 
adjacent 
to angle 
A 
Hypotenuse 
Hypotenuse  
Side opposite to angle A BC 
AC 
= 
Hypotenuse  
Side adjacent to angle A AB 
AC 
= 
Side opposite to angle A BC 
AB 
= 
Hypotenuse  
Side opposite to angle A 
AC 
BC 
= 
Hypotenuse  
Side adjacent to angle A 
AC 
AB 
= 
Side opposite to angle A 
AB 
BC 
= 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
 
Also, observe that tan A =                  and cot A =   
 
 
Trigonometric ratios of some specific angles: In this we will be calculating the 
trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and 
we will also develop the relationship between them.  
 
Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then 
the other angle is also 45°. ? ?A = ?C = 45° also AB = BC 
 
The sides in this triangle are in the ratio 1:1: v2 
 
Using the definition of trigonometric ratios, we have, 
 
Sin 45°  = 
BC
AC
= 
a
v2a
= 
1
v2
        Cosec 45°  = 
1
Sin 45°
= v2 
 
 
Cos 45° = 
AB
AC
= 
a
v2a
= 
1
v2
           Sec 45°  = 
1
Cos 45°
= v2 
 
Tan 45° = 
BC
AB
= 
a
a
=  1               Cot 45°  = 
1
Tan 45°
= 1 
 
 
Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 
30° then the other angle is 60°. Side opposite to 30° is always half of the 
hypotenuse; side opposite to 60° is always v3 times its adjacent side. 
 
?The sides in this triangle are in the ratio 1:2: v3 
 
Using the definition of trigonometric ratios, we have 
v2a 
a 
a 
Cos A  
Sin A  
Sin A  
Cos A  
45° 
45° 
C 
B 
A 
a 
v3a 
a 
2a 
30° 
60° 
C 
B 
A 
Page 3


 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Trigonometry: Trigonometry is a branch of mathematics that studies the 
relationships between the sides and angles of triangles. 
 
Trigonometric Ratios: The ratios of the sides of a right triangle are called 
trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are  the three 
common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) 
are the reciprocal of the ratios sin, cos and tan respectively. These are defined for 
acute angle A in right angled triangle ABC below: 
 
 
 
Sine of ?A =  
 
 
 
Cosine of ?A =  
 
 
 
Tangent of ?A=  
 
 
 
Cosecant of ?A= 
 
 
 
Secant of ?A = 
 
 
Cotangent of ?A =  
Side adjacent to angle A 
 
Side adjacent to angle A 
 
C 
B 
A 
Side 
opposite 
to angle 
A 
Side 
adjacent 
to angle 
A 
Hypotenuse 
Hypotenuse  
Side opposite to angle A BC 
AC 
= 
Hypotenuse  
Side adjacent to angle A AB 
AC 
= 
Side opposite to angle A BC 
AB 
= 
Hypotenuse  
Side opposite to angle A 
AC 
BC 
= 
Hypotenuse  
Side adjacent to angle A 
AC 
AB 
= 
Side opposite to angle A 
AB 
BC 
= 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
 
Also, observe that tan A =                  and cot A =   
 
 
Trigonometric ratios of some specific angles: In this we will be calculating the 
trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and 
we will also develop the relationship between them.  
 
Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then 
the other angle is also 45°. ? ?A = ?C = 45° also AB = BC 
 
The sides in this triangle are in the ratio 1:1: v2 
 
Using the definition of trigonometric ratios, we have, 
 
Sin 45°  = 
BC
AC
= 
a
v2a
= 
1
v2
        Cosec 45°  = 
1
Sin 45°
= v2 
 
 
Cos 45° = 
AB
AC
= 
a
v2a
= 
1
v2
           Sec 45°  = 
1
Cos 45°
= v2 
 
Tan 45° = 
BC
AB
= 
a
a
=  1               Cot 45°  = 
1
Tan 45°
= 1 
 
 
Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 
30° then the other angle is 60°. Side opposite to 30° is always half of the 
hypotenuse; side opposite to 60° is always v3 times its adjacent side. 
 
?The sides in this triangle are in the ratio 1:2: v3 
 
Using the definition of trigonometric ratios, we have 
v2a 
a 
a 
Cos A  
Sin A  
Sin A  
Cos A  
45° 
45° 
C 
B 
A 
a 
v3a 
a 
2a 
30° 
60° 
C 
B 
A 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Sin 30°  = 
BC
AC
= 
a
2a
= 
1
2
               Cosec 30°  = 
1
Sin 30°
= 2 
 
Cos 30° = 
AB
AC
= 
v3a
2a
= 
v3
2
           Sec 30°  = 
1
Cos 30°
=
2
v3
 
 
Tan 30° = 
BC
AB
= 
a
v3a
=
1
v3
            Cot 30°  = 
1
Tan 30°
= v3 
 
Similarly,  
           Sin 60°  = 
AB
AC
= 
v3a
2a
= 
v3
2
             Cosec 60°  = 
1
Sin 60°
=
2
v3
 
 
           Cos 60° = 
BC
AC
= 
a
2a
= 
1
2
                 Sec 60°     = 
1
Cos 60°
= 2 
 
           Tan 60° = 
AB
BC
= 
v3a
a
= v3             Cot 60°    = 
1
Tan 60°
=
1
v3
 
 
 
Trigonometric Ratios of 0° & 90° : if angle A is made smaller and smaller angle C 
becomes larger and larger. When angle C becomes smaller, side BC also decreases 
& finally when ?A becomes very close to 0°, AC will be almost the same as AB 
and side BC gets very close to zero. Therefore the value of sin A is very close to 0 
and cos A is very close to 1. 
Thus we have,  
 
  Sin 0°  = 0                               Cosec 0°  = 
1
Sin 0°
, which is not defined 
 
  Cos 0° = 1                               Sec 0°   = 
1
Cos 0°
= 1 
 
  Tan 0° = 
 Sin 0°  
Cos 0°
= 
0
1
= 0         Cot 60°    = 
1
Tan 0°
, which is again not defined. 
 
Now, if angle A is made larger and larger angle C becomes smaller and smaller. 
Therefore the length of side AB goes on decreasing. Point A gets closer to B. 
Finally when ?A is very close to 90°, ?C becomes very close to 0°. Side AC almost 
coincides with side BC so, sin A is very close to 1 and cos A is very close to 0.  
 
 
 
C 
B 
A 
Page 4


 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Trigonometry: Trigonometry is a branch of mathematics that studies the 
relationships between the sides and angles of triangles. 
 
Trigonometric Ratios: The ratios of the sides of a right triangle are called 
trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are  the three 
common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) 
are the reciprocal of the ratios sin, cos and tan respectively. These are defined for 
acute angle A in right angled triangle ABC below: 
 
 
 
Sine of ?A =  
 
 
 
Cosine of ?A =  
 
 
 
Tangent of ?A=  
 
 
 
Cosecant of ?A= 
 
 
 
Secant of ?A = 
 
 
Cotangent of ?A =  
Side adjacent to angle A 
 
Side adjacent to angle A 
 
C 
B 
A 
Side 
opposite 
to angle 
A 
Side 
adjacent 
to angle 
A 
Hypotenuse 
Hypotenuse  
Side opposite to angle A BC 
AC 
= 
Hypotenuse  
Side adjacent to angle A AB 
AC 
= 
Side opposite to angle A BC 
AB 
= 
Hypotenuse  
Side opposite to angle A 
AC 
BC 
= 
Hypotenuse  
Side adjacent to angle A 
AC 
AB 
= 
Side opposite to angle A 
AB 
BC 
= 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
 
Also, observe that tan A =                  and cot A =   
 
 
Trigonometric ratios of some specific angles: In this we will be calculating the 
trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and 
we will also develop the relationship between them.  
 
Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then 
the other angle is also 45°. ? ?A = ?C = 45° also AB = BC 
 
The sides in this triangle are in the ratio 1:1: v2 
 
Using the definition of trigonometric ratios, we have, 
 
Sin 45°  = 
BC
AC
= 
a
v2a
= 
1
v2
        Cosec 45°  = 
1
Sin 45°
= v2 
 
 
Cos 45° = 
AB
AC
= 
a
v2a
= 
1
v2
           Sec 45°  = 
1
Cos 45°
= v2 
 
Tan 45° = 
BC
AB
= 
a
a
=  1               Cot 45°  = 
1
Tan 45°
= 1 
 
 
Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 
30° then the other angle is 60°. Side opposite to 30° is always half of the 
hypotenuse; side opposite to 60° is always v3 times its adjacent side. 
 
?The sides in this triangle are in the ratio 1:2: v3 
 
Using the definition of trigonometric ratios, we have 
v2a 
a 
a 
Cos A  
Sin A  
Sin A  
Cos A  
45° 
45° 
C 
B 
A 
a 
v3a 
a 
2a 
30° 
60° 
C 
B 
A 
 
 
 
INTRODUCTION TO TRIGONOMETRY 
 
 
 
Sin 30°  = 
BC
AC
= 
a
2a
= 
1
2
               Cosec 30°  = 
1
Sin 30°
= 2 
 
Cos 30° = 
AB
AC
= 
v3a
2a
= 
v3
2
           Sec 30°  = 
1
Cos 30°
=
2
v3
 
 
Tan 30° = 
BC
AB
= 
a
v3a
=
1
v3
            Cot 30°  = 
1
Tan 30°
= v3 
 
Similarly,  
           Sin 60°  = 
AB
AC
= 
v3a
2a
= 
v3
2
             Cosec 60°  = 
1
Sin 60°
=
2
v3
 
 
           Cos 60° = 
BC
AC
= 
a
2a
= 
1
2
                 Sec 60°     = 
1
Cos 60°
= 2 
 
           Tan 60° = 
AB
BC
= 
v3a
a
= v3             Cot 60°    = 
1
Tan 60°
=
1
v3
 
 
 
Trigonometric Ratios of 0° & 90° : if angle A is made smaller and smaller angle C 
becomes larger and larger. When angle C becomes smaller, side BC also decreases 
& finally when ?A becomes very close to 0°, AC will be almost the same as AB 
and side BC gets very close to zero. Therefore the value of sin A is very close to 0 
and cos A is very close to 1. 
Thus we have,  
 
  Sin 0°  = 0                               Cosec 0°  = 
1
Sin 0°
, which is not defined 
 
  Cos 0° = 1                               Sec 0°   = 
1
Cos 0°
= 1 
 
  Tan 0° = 
 Sin 0°  
Cos 0°
= 
0
1
= 0         Cot 60°    = 
1
Tan 0°
, which is again not defined. 
 
Now, if angle A is made larger and larger angle C becomes smaller and smaller. 
Therefore the length of side AB goes on decreasing. Point A gets closer to B. 
Finally when ?A is very close to 90°, ?C becomes very close to 0°. Side AC almost 
coincides with side BC so, sin A is very close to 1 and cos A is very close to 0.  
 
 
 
C 
B 
A 
INTRODUCTION TO TRIGONOMETRY 
 
 
So we define sin90° = 1 & cos90° = 0, similarly other trigonometric ratios can be 
found. 
 
Trigonometric Ratios of Complementary angles: If the sum of two angles is one   
right angle or 90°, then one angle is said to be complementary of the other. Thus, ?° 
and (90 - ?)° are complementary to each other. 
 
sin (90°- A) = cos A ; cos (90°- A) = sin A 
tan (90°- A) = cot A; cot (90°- A) = tan A 
sec (90°- A) = csc A; csc (90°- A) = sec A                
These relations are valid for all the values of A lying between 0° and 90 
Trigonometric Identities: An equation is called an identity when it is true for all 
the value of the variables involved. Similarly, an equation involving trigonometric 
ratios of an angle is called a trigonometric identity, if it is true for all the values of 
the angle(s) involved.  
sin
2
???? + cos
2
???? = 1 
1 + tan
2
? = sec
-1
???? 
    1 + cot
2
? = cosec
-1
???? 
 
We obtain these identities by using Pythagoras theorem so these are also known as 
Pythagorean identities. 
?° 
(90- ?)° 
C 
B 
A 
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