Chapter 8 : Introduction to Trigonometry — Class 10 Mathematics NCERT Solutions | CBSE
Chapter 8 : Introduction to Trigonometry – NCERT Class 10 Mathematics Solutions | CBSE
Chapter 8 : Introduction to Trigonometry
8. Introduction to Trigonometry |
|
Exercise 8.1 Complete Solution Exercise 8.2 Complete Solution Exercise 8.3 Complete Solution Exercise 8.4 Complete Solution |
|
Trigonometric Ratios : (i) (iv) |
|
Formula : (a) |
|
Trigonometric Ratios of Complementary Angles (i) (iv) |
|
Trigonometric Identity : (a) |
EXERCISE 8.1
1. In 
, right-angled at B , 
. Determine :
(i) 
(ii) 
Solution : In figure :
Here ,
In 
, we have 
(i) We have, 
and 
(ii) We have, 
and 
2. In Fig. 8.13, find 
.
Fig. 8.13
Solution : Here, 
In 
, we have 
Now , 
and 
3. If 
, calculate 
and
.
Solution : Given, 
Let, 
In , we have
and 
4. Given 
, find
and
.
Solution : Given 
Let, 
In , we have
and 
5. Given 
, calculate all other trigonometric ratios .
Solution : Given,
Let, 
In , we have
, 
, 
, 
, 
and 
6. If 
and 
are acute angles such that
, then show that 
.
Solution : Given, and are acute angles of the triangle ABC .
Given , 
So, ABC is an isosceles triangle .
proved .
7. If 
, evaluate :
(i) 
(ii) 
Solution: Given , 
In given figure :
Let, 
In , we have 
(i) We have,
(ii) We have,
8. If 
, check whether 
or not .
Solution: Given ,
Let, 
In , we have
, 
and 
LHS: 
RHS : 
9. In triangle ABC, right-angled at B , if 
, find the value of :
(i) 
(ii) 
Solution : In given figure :
Given ,
Let, 
In , we have
, 
, 
and 
(i) We have, 
(ii) We have, 
10. In 
, right-angled at Q , 
and
. Determine the values of 
and 
.
Solution : Here , 
In , we have
, 
and 
11. State whether the following are true or false . Justify your answers :
(i) The value of 
is always less than 1 .
(ii) 
for some value of angle A.
(iii) 
is the abbreviation used for the cosecant of angle A .
(iv) 
is the product of cot and A .
(v) 
for some angles 
.
Solution: (i) False ,because the value of 
is always less than 1 .
[ Since, 
]
(ii) True , because the hypotenuse is the longest side in a right triangle, then the value of 
is always greater than or equal to 1 .
(iii) False, because is the abbreviation used for the cosine of angle A .
(iv) False , because 
is the ratio of base and perpendicular of the right triangle .
(v) False , because the hypotenuse is the longest side in a right triangle, then the value of 
is always less than 1 .
EXERCISE 8.2
1. Evaluate the following :
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution : (i) We have, 
(ii) We have, 
(iii) We have,
(iv) We have, 
(v) We have,
2. Choose the correct option and justify your choice :
(i) 
(A) 
(B) 
(C) 
(D) 
Solution: (A)
[ We have, 
]
(ii) 
(A) 
(B) 1 (C) 
(D) 0
Solution : (D) 0
[ We have, 
]
(iii) 
is true when 
(A) 0° (B) 30° (C) 45° (D) 60°
Solution : (A) 0°
[ Putting 
LHS : 
RHS : 
]
(iv) 
(A) 
(B) 
(C) (D)
Solution : (C) 
[ We have, 
]
3. If 
and 
,
find 
and
.
Solution : We have , 
and 
Putting 
in equation
, we have
Therefore , the value of A and B are 45° and 15° respectively .
(ii) If 
, 
and 
, 
then find 
and 
.
Solution: We have,
and
Putting 
in (i) , we get 
Therefore, the value of and 
.
4. State whether the following are true or false . Justify your answer .
(i) 
(ii) The value of 
increases as 
increases .
(iii) The value of 
increases as
increases .
(iv) 
for all values of .
(v) 
is not defined for 
.
Solution : (i) False , Because if you are putting the value of A and B then both sides are not equal .
(ii) True , because the value of 
increases as increases .
(iii) False , because the value of 
increases as increases .
[ i.e., the value of increases as decreases .]
(iv) False , only for 
is equal and other value of is not equal both sides .
(v) True, because the value of is not defined for .
EXERCISE 8.3
1. Evaluate : (i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) We have , 
(iii) We have , 
(iv) We have ,
2. Show that :
(i) 
(ii) 
Solution : (i) LHS : 
=1×1
=1
RHS
(ii) LHS : 
RHS
3. If 
where 
is an acute angle , find the value of
.
Solution : We have, 
Since, 
and 
are both acute angles .
Therefore, the value A is 36° .
4. If 
, prove that
.
Solution : Given, 
Since, 
and 
are both acute angles .
A=90°-B⇒A+B=90°
Proved .
5. If 
where
is an acute angle, find the value of A .
Solution : We have, 
Since, 
and 
are both acute angles .
Therefore, the value A is 22° .
6. If A , B and C are interior angles of a triangle ABC , then show that 
Solution : Since, A , B and C are interior angles of a triangle ABC respectively .
7. Express 
in terms of trigonometric ratios of angles between 0° and 45° .
Solution : We have ,
EXERCISE 8.4
1. Express the trigonometric ratios 
and 
in terms of .
Solution : We have , 
and 
2. Write all the other trigonometric ratios of 
in terms of .
Solution : We have , 
and 
3. Evaluate : (i) 
(ii) 
Solution : (i) We have ,
(ii) We have,
4. Choose the correct option . Justify your choice .
(i) 
(A) 1 (B) 9 (C) 8 (D) 0
Solution : (B) 9
[ We have, 
]
(ii) 
(A) 0 (B) 1 (C) 2 (D) 
Solution : (C) 2
[ We have ,
]
(iii) 
(A) 
(B) 
(C) 
(D) 
Solution : (D) 
[ We have, 
]
(iv) 
(A) 
(B) (C) 
(D) 
Solution : (D)
[ We have , 
]
5. Prove the following identities , where the angles involved are acute angles for which the expressions are defined .
(i) 
Solution : L.H.S : 
R.H.S
(ii) 
Solution : L.H.S. : 
R.H.S. Proved.
(iii) 
[Hint : Write the expression in terms of 
and 
]
Solution : L.H.S : 
R.H.S
(iv) 
[Hint : Simplify LHS and RHS separately]
Solution : L.H.S : 
RHS : 
(v) 
, using the identity 
.
Solution: LHS : 
RHS
(vi) 
Solution : LHS : 
(vii) 
Solution : L.H.S. : 
R.H.S.
(viii) 
Solution : L.H.S. : 
R.H.S Proved.
(ix) 
Solution : LHS : 
RHS : 
LHS = RHS
(x) 
Solution : First part : 
Second part : 
First part = Second part = Third part Proved.
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