**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 :

**Answer:**

We have,

In

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

So,

and

**Q2 **In Fig. 8.13, find .

**Answer:**

We have,

So, by using Pythagoras theorem,

Now, According to question,

= 5/12 – 5/12 = 0

**Q3 **If calculate and .

**Answer:**

Suppose

So,

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

Therefore,

**Q4 **Given find and .

**Answer:**

We have,

It implies that In the triangle ABC in which

Now, by using Pythagoras theorem,

So,

and

**Q5 **Given calculate all other trigonometric ratios.

**Answer:**

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

BC = 5 unit

Therefore,

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

**Answer:**

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

Therefore,

**Q7 **If evaluate:

**Answer:**

Given that,

Draw a right-angled triangle ABC in which

Now, By using Pythagoras theorem,

So,

and

**Q8 **If check wether or not.

**Answer:**

Given that,

ABC is a right-angled triangle in which

By using Pythagoras theorem, In triangle ABC,

AC = 5 units

So,

Put the values of above trigonometric ratios, we get;

LHS

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

**Answer:**

Given a triangle ABC, right-angled at B and

By using Pythagoras theorem,

AC = 2

Now,

Therefore,

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

**Answer:**

PQ = 5 cm

and

According to question,

In triangle

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

(ii)

(iii)

(iv)

(v)

**Answer:**

(i) False,

because

(ii) TRue,

because

(iii) False,

Because

(iv) False,

because the term

(v) False,

because

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

**Q1 **Evaluate the following :

**Answer:**

As we know,

the value of

**Q1 **Evaluate the following :

**Answer:**

We know the value of

According to question,

**Q1 **Evaluate the following :

**Answer:**

we know the value of

After putting these values

**Q1 **Evaluate the following :

**Answer:**

It is known that the values of the given trigonometric functions,

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

**Q1 **Evaluate the following :

**Answer:**

We know the values of-

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

**Q2 **Choose the correct option and justify your choice :

**Answer:**

Put the value of **tan 30 **in the given question-

The correct option is (A)

**Q2 **Choose the correct option and justify your choice :

**Answer:**

The correct option is (D)

We know that

So,

**Q2 **Choose the correct option and justify your choice :

**Answer:**

The correct option is (A)

We know that

So,

**Q2 **Choose the correct option and justify your choice :

**Answer:**

Put the value of

The correct option is (C)

**Q3 **If and find

**Answer:**

Given that,

So,

therefore,

By solving the equation (i) and (ii) we get;

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

**Answer:**

(i) False,

Let A = B =

Then,

(ii) True,

Take

whent

(iii) False,

(iv) False,

Let

(v) True,

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

**Q1 **Evaluate :

**Answer:**

We can write the above equation as;

By using the identity of

Therefore,

So, the answer is 1.

**Q1 **Evaluate :

**Answer:**

The above equation can be written as ;

It is known that,

Therefore, equation (i) becomes,

So, the answer is 1.

**Q1 **Evaluate :

**Answer:**

The above equation can be written as ;

It is known that

Therefore, equation (i) becomes,

So, the answer is 0.

**Q1 **Evaluate :

**Answer:**

This equation can be written as;

We know that

Therefore, equation (i) becomes;

So, the answer is 0.

**Q2 **Show that :

**Answer:**

Taking Left Hand Side (LHS)

=

Hence proved.

**Q2 **Show that :

**Answer:**

Taking Left Hand Side (LHS)

=

=

=

= 0

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

**Answer:**

We have,

we know that,

**Q4 **If , prove that .

**Answer:**

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 .

**Answer:**

We have,

According to question,

We know that

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

**Answer:**

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 .

**Answer:**

By using the identity of

We know that,

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 .

**Answer:**

We know that

(i)

(ii) We know the identity of

(iii)

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

**Answer:**

We know that the identity

**Q3 **Evaluate :

**Answer:**

The above equation can be written as;

(Since

**Q3 **Evaluate :

**Answer:**

**We know that**

**Therefore,**

**Q4 **Choose the correct option. Justify your choice.

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

**Answer:**

The correct option is (B) = 9

and it is known that sec^{2}A-tan^{2}A=1

Therefore, equation (i) becomes,

**Q4 **Choose the correct option. Justify your choice.

(A) 0 (B) 1 (C) 2 (D) –1

**Answer:**

The correct option is (C)

we can write his above equation as;

= 2

**Q4 **Choose the correct option. Justify your choice.

**Answer:**

The correct option is (D)

**Q4 **Choose the correct option. Justify your choice.

**Answer:**

The correct option is (D)

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.

**Answer:**

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.

**Answer:**

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

**Answer:**

We need to prove-

Taking LHS;

By using the identity a ^{3 }– b ^{3 }=(a – b) (a ^{2 }+ b ^{2 }+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]

### Answer:

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

**Answer:**

We need to prove –

Dividing the numerator and denominator by

Hence Proved.

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

**Answer:**

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.

**Answer:**

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.

**Answer:**

Given equation,

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]

**Answer:**

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.

**Answer:**

We need to prove,

Taking LHS;

Taking RHS;

LHS = RHS

Hence proved.