Textbook: Trigonometry | Mathematics Class 10 (Maharashtra SSC Board) PDF Download

 Page 1


124
sin q =  , cos q =   ,
2. Complete the relations in ratios given below .
 (i) 
sin
cos
q
q
 =    (ii) sin q =  cos (90 - ) 
 (iii) cos q = sin (90 - ) (iv) tan q x tan (90 - q) =  
3. Complete the equation. 
 sin
2
 q + cos
2
 q =  
4. Write the values of the following trigonometric ratios. 
 (i) sin30° = 
1
 (ii) cos30° =  (iii) tan30° = 
 (iv) sin60° =  (v) cos45° =  (vi) tan45° =    
      In std IX, we have studied some trigonometric ratios of some acute angles. 
                Now we  are going to study some more trigonometric ratios of acute angles.
A
B C
q
Fig. 6.1
tan q =  
· Trigonometric ratios      · Trigonometric identities
· Angle of elevation and angle of depression · Problems based on heights and  
            distances
Let’s recall.
1. Fill in the blanks with reference to figure 6.1 .
Let’s study.
6
Trigonometry
Page 2


124
sin q =  , cos q =   ,
2. Complete the relations in ratios given below .
 (i) 
sin
cos
q
q
 =    (ii) sin q =  cos (90 - ) 
 (iii) cos q = sin (90 - ) (iv) tan q x tan (90 - q) =  
3. Complete the equation. 
 sin
2
 q + cos
2
 q =  
4. Write the values of the following trigonometric ratios. 
 (i) sin30° = 
1
 (ii) cos30° =  (iii) tan30° = 
 (iv) sin60° =  (v) cos45° =  (vi) tan45° =    
      In std IX, we have studied some trigonometric ratios of some acute angles. 
                Now we  are going to study some more trigonometric ratios of acute angles.
A
B C
q
Fig. 6.1
tan q =  
· Trigonometric ratios      · Trigonometric identities
· Angle of elevation and angle of depression · Problems based on heights and  
            distances
Let’s recall.
1. Fill in the blanks with reference to figure 6.1 .
Let’s study.
6
Trigonometry
125
A
B
C
q
Fig. 6.2
      In figure 6.2, 
  sinq = 
AB
AC 
 
\  cosecq = 
1
sinq
 
   = 
1
AB
AC
   
   = 
AC
AB 
 
It means, 
cosecq = 
hypotenuse
opposite side
          
   tanq = 
AB
BC 
 
\  cot q = 
1
tanq
 
    = 
1
AB
BC
 cot q = 
BC
AB 
 = 
adjacent side
opposite side
  cosq = 
BC
AC 
  secq = 
1
cosq
   = 
1
BC
AC
   = 
AC
BC 
It means,
 secq = 
hypotenuse
adjacent side
 
You know that, 
 tanq =
sinq
cosq
 
\ cot q = 
1
tanq
   = 
1
sin
cos
q
q
   =
cos q
sin q
\ cot q =
cos q
sin q
Let’s learn.
cosec, sec and cot ratios
Multiplicative inverse or the reciprocal of sine ratio is called cosecant ratio. It is 
written in brief as cosec. \ cosecq = 
1
sinq
Similarly, multiplicative inverses or reciprocals of cosine and tangent ratios are 
called “secant” and  “cotangent” ratios respectively. They are written in brief as sec 
and cot.
\ secq = 
1
cosq
  and  cotq = 
1
tanq
 
Page 3


124
sin q =  , cos q =   ,
2. Complete the relations in ratios given below .
 (i) 
sin
cos
q
q
 =    (ii) sin q =  cos (90 - ) 
 (iii) cos q = sin (90 - ) (iv) tan q x tan (90 - q) =  
3. Complete the equation. 
 sin
2
 q + cos
2
 q =  
4. Write the values of the following trigonometric ratios. 
 (i) sin30° = 
1
 (ii) cos30° =  (iii) tan30° = 
 (iv) sin60° =  (v) cos45° =  (vi) tan45° =    
      In std IX, we have studied some trigonometric ratios of some acute angles. 
                Now we  are going to study some more trigonometric ratios of acute angles.
A
B C
q
Fig. 6.1
tan q =  
· Trigonometric ratios      · Trigonometric identities
· Angle of elevation and angle of depression · Problems based on heights and  
            distances
Let’s recall.
1. Fill in the blanks with reference to figure 6.1 .
Let’s study.
6
Trigonometry
125
A
B
C
q
Fig. 6.2
      In figure 6.2, 
  sinq = 
AB
AC 
 
\  cosecq = 
1
sinq
 
   = 
1
AB
AC
   
   = 
AC
AB 
 
It means, 
cosecq = 
hypotenuse
opposite side
          
   tanq = 
AB
BC 
 
\  cot q = 
1
tanq
 
    = 
1
AB
BC
 cot q = 
BC
AB 
 = 
adjacent side
opposite side
  cosq = 
BC
AC 
  secq = 
1
cosq
   = 
1
BC
AC
   = 
AC
BC 
It means,
 secq = 
hypotenuse
adjacent side
 
You know that, 
 tanq =
sinq
cosq
 
\ cot q = 
1
tanq
   = 
1
sin
cos
q
q
   =
cos q
sin q
\ cot q =
cos q
sin q
Let’s learn.
cosec, sec and cot ratios
Multiplicative inverse or the reciprocal of sine ratio is called cosecant ratio. It is 
written in brief as cosec. \ cosecq = 
1
sinq
Similarly, multiplicative inverses or reciprocals of cosine and tangent ratios are 
called “secant” and  “cotangent” ratios respectively. They are written in brief as sec 
and cot.
\ secq = 
1
cosq
  and  cotq = 
1
tanq
 
126
For more information :
 The great Indian mathematician Aryabhata was born 
in 476 A.D. in Kusumpur which was near Patna in Bihar. 
He has done important work in Arithmetic, Algebra and 
Geometry. In the book ‘Aryabhatiya’ he has written many 
mathematical formulae. For example, 
(1) In an Arithmetic Progression, formulae for n
th
 term and  
      the sum of first n terms.
(2) The formula to approximate 
2
 
(3) The correct value of  p upto four decimals, p = 3.1416.
 In the study of Astronomy he used trigonometry and 
the sine ratio of an angle for the first time. 
 Comparing with the mathematics in the rest of the world at that time, his 
work was great and was studied all over India and was carried to Europe through 
Middle East. 
 Most observers at that time believed that the earth is immovable and the 
Sun,  the Moon and stars move arround the earth. But Aryabhata noted that when 
we travel in a boat on the river, objects like trees, houses on the bank appear to 
move in the opposite direction. ‘Similarly’, he said ‘the Sun, Moon and the stars 
are observed by people on the earth to be moving in the opposite direction while 
in reality the Earth moves !’
 On 19 April 1975, India sent the first satellite in the space and it was named 
‘Aryabhata’ to commemorate the great Mathematician of India.
  
Remember this !
The relation between the trigonometric ratios,
according to the definitions of cosec, sec and cot ratios
· 
1
sinq
 = cosec q  \ sin q ´ cosec q = 1
· 
1
cosq
 = sec q  \ cos q ´ sec q = 1
· 
1
tanq
 = cot q  \ tan q ´ cot q = 1
Page 4


124
sin q =  , cos q =   ,
2. Complete the relations in ratios given below .
 (i) 
sin
cos
q
q
 =    (ii) sin q =  cos (90 - ) 
 (iii) cos q = sin (90 - ) (iv) tan q x tan (90 - q) =  
3. Complete the equation. 
 sin
2
 q + cos
2
 q =  
4. Write the values of the following trigonometric ratios. 
 (i) sin30° = 
1
 (ii) cos30° =  (iii) tan30° = 
 (iv) sin60° =  (v) cos45° =  (vi) tan45° =    
      In std IX, we have studied some trigonometric ratios of some acute angles. 
                Now we  are going to study some more trigonometric ratios of acute angles.
A
B C
q
Fig. 6.1
tan q =  
· Trigonometric ratios      · Trigonometric identities
· Angle of elevation and angle of depression · Problems based on heights and  
            distances
Let’s recall.
1. Fill in the blanks with reference to figure 6.1 .
Let’s study.
6
Trigonometry
125
A
B
C
q
Fig. 6.2
      In figure 6.2, 
  sinq = 
AB
AC 
 
\  cosecq = 
1
sinq
 
   = 
1
AB
AC
   
   = 
AC
AB 
 
It means, 
cosecq = 
hypotenuse
opposite side
          
   tanq = 
AB
BC 
 
\  cot q = 
1
tanq
 
    = 
1
AB
BC
 cot q = 
BC
AB 
 = 
adjacent side
opposite side
  cosq = 
BC
AC 
  secq = 
1
cosq
   = 
1
BC
AC
   = 
AC
BC 
It means,
 secq = 
hypotenuse
adjacent side
 
You know that, 
 tanq =
sinq
cosq
 
\ cot q = 
1
tanq
   = 
1
sin
cos
q
q
   =
cos q
sin q
\ cot q =
cos q
sin q
Let’s learn.
cosec, sec and cot ratios
Multiplicative inverse or the reciprocal of sine ratio is called cosecant ratio. It is 
written in brief as cosec. \ cosecq = 
1
sinq
Similarly, multiplicative inverses or reciprocals of cosine and tangent ratios are 
called “secant” and  “cotangent” ratios respectively. They are written in brief as sec 
and cot.
\ secq = 
1
cosq
  and  cotq = 
1
tanq
 
126
For more information :
 The great Indian mathematician Aryabhata was born 
in 476 A.D. in Kusumpur which was near Patna in Bihar. 
He has done important work in Arithmetic, Algebra and 
Geometry. In the book ‘Aryabhatiya’ he has written many 
mathematical formulae. For example, 
(1) In an Arithmetic Progression, formulae for n
th
 term and  
      the sum of first n terms.
(2) The formula to approximate 
2
 
(3) The correct value of  p upto four decimals, p = 3.1416.
 In the study of Astronomy he used trigonometry and 
the sine ratio of an angle for the first time. 
 Comparing with the mathematics in the rest of the world at that time, his 
work was great and was studied all over India and was carried to Europe through 
Middle East. 
 Most observers at that time believed that the earth is immovable and the 
Sun,  the Moon and stars move arround the earth. But Aryabhata noted that when 
we travel in a boat on the river, objects like trees, houses on the bank appear to 
move in the opposite direction. ‘Similarly’, he said ‘the Sun, Moon and the stars 
are observed by people on the earth to be moving in the opposite direction while 
in reality the Earth moves !’
 On 19 April 1975, India sent the first satellite in the space and it was named 
‘Aryabhata’ to commemorate the great Mathematician of India.
  
Remember this !
The relation between the trigonometric ratios,
according to the definitions of cosec, sec and cot ratios
· 
1
sinq
 = cosec q  \ sin q ´ cosec q = 1
· 
1
cosq
 = sec q  \ cos q ´ sec q = 1
· 
1
tanq
 = cot q  \ tan q ´ cot q = 1
127
A B
C
q
Fig. 6.3
* The table of the values of  trigonometric ratios of angles 0°,30°,45°,60° and  90° .
Trigonometric 
ratio
Angle (q)
0° 30° 45° 60° 90°
sin q 0
1
2
1
2
3
2
1
cos q 1
3
2
1
2
1
2
0
tan q 0
1
3
1 3 Not defined
cosec q
= 
1
sin q
Not defined
2 2
2
3
1
sec q
= 
1
cos q
1
2
3
2 2
Not defined
cot q
= 
1
tan q
Not defined 3 1
1
3
0
Let’s learn.
Trigonometric identities
  In the figure 6.3, D ABC is a right angled triangle, ÐB= 90°
  (i) sinq = 
BC
AC 
  (ii) cosq = 
AB
AC 
   
  (iii) tanq = 
BC
AB 
  (iv) cosecq = 
AC
BC 
  
  (v) secq = 
AC
AB 
  (vi) cotq =  
AB
BC
   
  By Pythagoras therom,  
  BC
2
 + AB
2
 = AC
2
 . . . . .(I)
  Dividing both the sides of (1) by AC
2
 
  
BC
2
+ AB
2
AC
2
 = 
AC
2
AC
2
 
Page 5


124
sin q =  , cos q =   ,
2. Complete the relations in ratios given below .
 (i) 
sin
cos
q
q
 =    (ii) sin q =  cos (90 - ) 
 (iii) cos q = sin (90 - ) (iv) tan q x tan (90 - q) =  
3. Complete the equation. 
 sin
2
 q + cos
2
 q =  
4. Write the values of the following trigonometric ratios. 
 (i) sin30° = 
1
 (ii) cos30° =  (iii) tan30° = 
 (iv) sin60° =  (v) cos45° =  (vi) tan45° =    
      In std IX, we have studied some trigonometric ratios of some acute angles. 
                Now we  are going to study some more trigonometric ratios of acute angles.
A
B C
q
Fig. 6.1
tan q =  
· Trigonometric ratios      · Trigonometric identities
· Angle of elevation and angle of depression · Problems based on heights and  
            distances
Let’s recall.
1. Fill in the blanks with reference to figure 6.1 .
Let’s study.
6
Trigonometry
125
A
B
C
q
Fig. 6.2
      In figure 6.2, 
  sinq = 
AB
AC 
 
\  cosecq = 
1
sinq
 
   = 
1
AB
AC
   
   = 
AC
AB 
 
It means, 
cosecq = 
hypotenuse
opposite side
          
   tanq = 
AB
BC 
 
\  cot q = 
1
tanq
 
    = 
1
AB
BC
 cot q = 
BC
AB 
 = 
adjacent side
opposite side
  cosq = 
BC
AC 
  secq = 
1
cosq
   = 
1
BC
AC
   = 
AC
BC 
It means,
 secq = 
hypotenuse
adjacent side
 
You know that, 
 tanq =
sinq
cosq
 
\ cot q = 
1
tanq
   = 
1
sin
cos
q
q
   =
cos q
sin q
\ cot q =
cos q
sin q
Let’s learn.
cosec, sec and cot ratios
Multiplicative inverse or the reciprocal of sine ratio is called cosecant ratio. It is 
written in brief as cosec. \ cosecq = 
1
sinq
Similarly, multiplicative inverses or reciprocals of cosine and tangent ratios are 
called “secant” and  “cotangent” ratios respectively. They are written in brief as sec 
and cot.
\ secq = 
1
cosq
  and  cotq = 
1
tanq
 
126
For more information :
 The great Indian mathematician Aryabhata was born 
in 476 A.D. in Kusumpur which was near Patna in Bihar. 
He has done important work in Arithmetic, Algebra and 
Geometry. In the book ‘Aryabhatiya’ he has written many 
mathematical formulae. For example, 
(1) In an Arithmetic Progression, formulae for n
th
 term and  
      the sum of first n terms.
(2) The formula to approximate 
2
 
(3) The correct value of  p upto four decimals, p = 3.1416.
 In the study of Astronomy he used trigonometry and 
the sine ratio of an angle for the first time. 
 Comparing with the mathematics in the rest of the world at that time, his 
work was great and was studied all over India and was carried to Europe through 
Middle East. 
 Most observers at that time believed that the earth is immovable and the 
Sun,  the Moon and stars move arround the earth. But Aryabhata noted that when 
we travel in a boat on the river, objects like trees, houses on the bank appear to 
move in the opposite direction. ‘Similarly’, he said ‘the Sun, Moon and the stars 
are observed by people on the earth to be moving in the opposite direction while 
in reality the Earth moves !’
 On 19 April 1975, India sent the first satellite in the space and it was named 
‘Aryabhata’ to commemorate the great Mathematician of India.
  
Remember this !
The relation between the trigonometric ratios,
according to the definitions of cosec, sec and cot ratios
· 
1
sinq
 = cosec q  \ sin q ´ cosec q = 1
· 
1
cosq
 = sec q  \ cos q ´ sec q = 1
· 
1
tanq
 = cot q  \ tan q ´ cot q = 1
127
A B
C
q
Fig. 6.3
* The table of the values of  trigonometric ratios of angles 0°,30°,45°,60° and  90° .
Trigonometric 
ratio
Angle (q)
0° 30° 45° 60° 90°
sin q 0
1
2
1
2
3
2
1
cos q 1
3
2
1
2
1
2
0
tan q 0
1
3
1 3 Not defined
cosec q
= 
1
sin q
Not defined
2 2
2
3
1
sec q
= 
1
cos q
1
2
3
2 2
Not defined
cot q
= 
1
tan q
Not defined 3 1
1
3
0
Let’s learn.
Trigonometric identities
  In the figure 6.3, D ABC is a right angled triangle, ÐB= 90°
  (i) sinq = 
BC
AC 
  (ii) cosq = 
AB
AC 
   
  (iii) tanq = 
BC
AB 
  (iv) cosecq = 
AC
BC 
  
  (v) secq = 
AC
AB 
  (vi) cotq =  
AB
BC
   
  By Pythagoras therom,  
  BC
2
 + AB
2
 = AC
2
 . . . . .(I)
  Dividing both the sides of (1) by AC
2
 
  
BC
2
+ AB
2
AC
2
 = 
AC
2
AC
2
 
128
 \  
BC
2
AC
2
 
 + 
AB
2
AC
2
 
 = 1
 \  BC
AC
AB
AC
?
?
?
?
?
?
+
?
?
?
?
?
?
=
22
1
 \(sinq)
2
 + (cosq)
2
 = 1 .... [(sinq)
2
 is written as sin
2
q and (cosq)
2
 is written
               as cos
2
q.]
  sin
2 
q + cos
2 
q = 1 .......... (II)
  Now dividing both the sides of  equation (II)  by sin
2
q     
  
sin
sin
cos
sinsin
2
2
2
22
1 ?
?
?
??
+=
 
  1 + cot
2 
q = cosec
2 
q .......... (III)
  Dividing both the sides of equation  (II) by cos
2
q   
  
sin
cos
cos
coscos
2
2
2
22
1 ?
?
?
??
+=
   tan
2 
q + 1 = sec
2 
q 
  1 + tan
2 
q = sec
2 
q .......... (IV)
  Relations (II),(III), and (IV) are fundamental trigonometric identities.
?????????????? Solved Examples ?????????????
Ex. (1)  If sinq = 
20
29
 then find cosq  
Solution ? Method I 
  We have
  sin
2 
q + cos
2 
q = 1
  
20
29
2
?
?
?
?
?
?
 + cos
2 
q = 1
  
400
841
 + cos
2 
q = 1
  cos
2 
q = 1 - 
400
841
            = 
441
841
 
 Taking square root of   
 both sides.
             cosq = 
21
29
 
Method II 
sinq = 
20
29
 
from figure, sinq = 
AB
AC 
 
\ AB = 20k  and AC = 29k
     Let BC = x.
According to Pythagoras therom,
 AB
2
+ BC
2
 = AC
2
                       
(20k)
2
+ x
2
 = (29k)
2
 
400k
2
+ x
2
 = 841k
2
 
 x
2
 = 841k
2 
- 400k
2
 
     = 441k
2
              
\   x = 21k
 
        \ cos
 
q = 
BC
AC 
 = 
21k
29k 
 = 
21
29 
 
Fig. 6.4
20k
29k A
B C
x
q