# NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry

##### NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry

NCERT Chapter 8 explains the relationship between the angles and sides of a right triangle. NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry are required for students taking the Class 10 Board exams. These NCERT solutions for Class 10 Maths chapter 8 are strictly based on the NCERT books for Class 10 Maths. NCERT Class 10 maths solutions chapter 8 provides a thorough explanation of each question in the textbook. Furthermore, NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry provides many tips and tricks for answering questions easily. You can also look through the NCERT solutions for Class 10 for other subjects.

## NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry Excercise: 8.1

### Q1 In  , right-angled at  ,  . Determine :

We have,
In  ,  B = 90, and the length of the base (AB) = 24 cm and length of perpendicular (BC) = 7 cm
So, by using Pythagoras theorem,

Therefore,

AC = 25 cm

Now,
(i)

(ii) For angle C, AB is perpendicular to the base (BC). Here B indicates to Base and P means perpendicular wrt angle  C
So,
and

Q2 In Fig. 8.13, find  .

We have,  PQR is a right-angled triangle, length of PQ and PR are 12 cm and 13 cm respectively.
So, by using Pythagoras theorem,

Now, According to question,
=
= 5/12 – 5/12 = 0

Q3 If  calculate  and  .

Suppose  ABC is a right-angled triangle in which  and we have
So,

Let the length of AB be 4 unit and the length of BC = 3 unit So, by using Pythagoras theorem,
units
Therefore,
and

Q4 Given  find  and  .

We have,

It implies that In the triangle ABC in which  . The length of AB be 8 units and the length of BC = 15 units

Now, by using Pythagoras theorem,

units

So,
and

Q5 Given  calculate all other trigonometric ratios.

We have,

It means the Hypotenuse of the triangle is 13 units and the base is 12 units.
Let ABC is a right-angled triangle in which  B is 90 and AB is the base, BC is perpendicular height and AC is the hypotenuse.

By using Pythagoras theorem,

BC = 5 unit

Therefore,

Q6 If  and  are acute angles such that  , then show that  .

We have, A and B are two acute angles of triangle ABC and

According to question, In triangle ABC,

Therefore,  A =  B [angle opposite to equal sides are equal]

Q7 If  evaluate:

Given that,

perpendicular (AB) = 8 units and Base (AB) = 7 units
Draw a right-angled triangle ABC in which
Now, By using Pythagoras theorem,

So,
and

Q8 If  check wether  or not.

Given that,

ABC is a right-angled triangle in which  and the length of the base AB is 4 units and length of perpendicular is 3 units

By using Pythagoras theorem, In triangle ABC,

AC = 5 units

So,

Put the values of above trigonometric ratios, we get;

LHS  RHS

Q9 In triangle  , right-angled at  , if  find the value of:

Given a triangle ABC, right-angled at B and

According to question,
By using Pythagoras theorem,

AC = 2
Now,

Therefore,

Q10 In  , right-angled at  ,  and  . Determine the values of

We have, PR + QR = 25 cm………….(i)
PQ = 5 cm
and
According to question,
In triangle  PQR,
By using Pythagoras theorem,

PR – QR = 1……..(ii)

From equation(i) and equation(ii), we get;
PR = 13 cm and QR = 12 cm.

therefore,

Q11 State whether the following are true or false. Justify your answer.

(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 angle

(i) False,
because  , which is greater than 1

(ii) TRue,
because

(iii) False,
Because  abbreviation is used for cosine A.

(iv) False,
because the term  is a single term, not a product.

(v) False,
because  lies between (-1 to +1) [  ]

## NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry Excercise: 8.2

Q1 Evaluate the following :

As we know,
the value of  ,

Q1 Evaluate the following :

We know the value of

and

According to question,

Q1 Evaluate the following :

we know the value of

,  and  ,

After putting these values

Q1 Evaluate the following :

………………(i)
It is known that the values of the given trigonometric functions,

Put all these values in equation (i), we get;

Q1 Evaluate the following :

…………………(i)
We know the values of-

By substituting all these values in equation(i), we get;

Q2 Choose the correct option and justify your choice :

Put the value of tan 30 in the given question-

The correct option is (A)

Q2 Choose the correct option and justify your choice :

The correct option is (D)

We know that
So,

Q2 Choose the correct option and justify your choice :

is true when  =

The correct option is (A)

We know that
So,

Q2 Choose the correct option and justify your choice :

Put the value of

The correct option is (C)

### Q3 If  and   find

Given that,

So,  ……….(i)

therefore,  …….(ii)
By solving the equation (i) and (ii) we get;

and

Q4 State whether the following are true or false. Justify your answer.

The value of  increases as  increases.
The value of  increases as  increases.
for all values of  .
is not defined for

(i) False,
Let A = B =
Then,

(ii) True,
Take
whent
= 0 then zero(0),
= 30 then value of  is 1/2 = 0.5
= 45 then value of  is 0.707

(iii) False,

(iv) False,
Let  = 0

(v) True,
(not defined)

## NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry Excercise: 8.3

Q1 Evaluate :

We can write the above equation as;

By using the identity of
Therefore,

Q1 Evaluate :

The above equation can be written as ;

………(i)
It is known that,
Therefore, equation (i) becomes,

Q1 Evaluate :

The above equation can be written as ;
………………..(i)
It is known that
Therefore, equation (i) becomes,

Q1 Evaluate :

This equation can be written as;
……………..(i)
We know that

Therefore, equation (i) becomes;
= 0

Q2 Show that :

Taking Left Hand Side (LHS)

[it is known that  and

Hence proved.

Q2 Show that :

Taking Left Hand Side (LHS)

[it is known that  and  ]
= 0

Q3 If  , where  is an acute angle, find the value of  .

We have,
2A =  (A –  )
we know that,

Q4 If  , prove that  .

We have,

and we know that
therefore,

A = 90 – B
A + B = 90
Hence proved.

Q5 If  , where  is an acute angle, find the value of  .

We have,
, Here 4A is an acute angle
According to question,
We know that

Q6 If  and  are interior angles of a triangle  , then show that

Given that,
A, B and C are interior angles of
To prove –

Now,
In triangle  ,
A + B + C =

Hence proved.

Q7 Express  in terms of trigonometric ratios of angles between  and  .

By using the identity of  and

We know that,
and
the above equation can be written as;

## NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Excercise: 8.4

Q1 Express the trigonometric ratios  and  in terms of  .

We know that
(i)

(ii) We know the identity of

(iii)

Q2 Write all the other trigonometric ratios of  in terms of  .

We know that the identity

Q3 Evaluate :

………………..(i)

The above equation can be written as;

(Since  )

We know that

Therefore,

Q4 Choose the correct option. Justify your choice.

(A) 1 (B) 9 (C) 8 (D) 0

The correct option is (B) = 9

………….(i)

and it is known that sec2A-tan2A=1

Therefore, equation (i) becomes,

Q4 Choose the correct option. Justify your choice.

(A) 0 (B) 1 (C) 2 (D) –1

The correct option is (C)

…………………..(i)

we can write his above equation as;

= 2

Q4 Choose the correct option. Justify your choice.

The correct option is (D)

Q4 Choose the correct option. Justify your choice.

The correct option is (D)

……………………..eq (i)

The above equation can be written as;

We know that

therefore,

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

We need to prove-

Now, taking LHS,

LHS = RHS

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

We need to prove-

taking LHS;

= RHS

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Hint: Write the expression in terms of  and  ]

We need to prove-

Taking LHS;

By using the identity a – b =(a – b) (a + b +ab)

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Hint : Simplify LHS and RHS separately]

We need to prove-

taking LHS;

Taking RHS;
We know that identity

LHS = RHS

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.  , using the identity

We need to prove –

Dividing the numerator and denominator by  , we get;

Hence Proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

We need to prove –

Taking LHS;
By rationalising the denominator, we get;

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

We need to prove –

Taking LHS;
[we know the identity  ]

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Given equation,
………………(i)

Taking LHS;

[since  ]

Hence proved

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Hint : Simplify LHS and RHS separately]

We need to prove-

Taking LHS;

Taking RHS;

LHS = RHS

Hence proved.

Q5 Prove the following identities, where the angles involved are acute angles for which the expressions are defined.