NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry

NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry Solutions, NCERT Solutions For Class 10 Maths, CBSE Solutions For Class 10 Mathematics to each chapter is provided in the list so that you can easily browse throughout different chapter NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry Notes and select needs one.

NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry

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Also, you can read the CBSE book online in these sections Solutions by Expert Teachers as per (CBSE) Book guidelines. NCERT Class 10 Mathematics Textual Question Answer. These solutions are part of NCERT All Subject Solutions. Here we have given NCERT Class 10 Mathematics Chapter 8 Introduction to Trigonometry Solutions for All Subject, You can practice these here.

Introduction to Trigonometry

Chapter – 8

Exercise 8.1

1. In ∆ABC right angled at B, AB = 24 cm, BC = 7 m. Determine.

2. In the given figure find tan P − cot R

3. If sin A =3/4 calculate cos A and tan A.

Ans: 

4. Given 15 cot A = 8. Find sin A and sec A

Ans; Consider a right-angled triangle, right-angled at B.  

5. Given sec θ = ,13/12 calculate all other trigonometric ratios.

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Ans:  

If AC is 13k, AB will be 12k, where k is a positive integer.

Applying Pythagoras theorem in ∆ABC, we obtain

(AC)2 = (AB)2 + (BC)2

(13k)2 = (12k)2 + (BC)2

169k2 = 144k2 + BC2

25k2 = BC2

BC = 5k 

6. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

Ans: Let us consider a triangle ABC in which CD ⊥ AB. 

It is given that cos A = cos B

AB/AC = BD/BC …………………. (1)

We have to prove ∠A = ∠B. To prove this, let us extend AC to P such that BC = CP.

7. If cot θ = ⅞  evaluate

Ans: Let us consider a right triangle ABC, right-angled at point B.

If BC is 7k, then AB will be 8k, where k is a positive integer. Applying Pythagoras theorem in ∆ABC, we obtain  

AC2 = AB2 + BC2

= (8k)2 + (7k)2

= 64k2 + 49k2

= 113k2 

Ans: Do yourself.

8.

Ans: If 3 cot A = 4,

It is given that 3 cot = ¾   A = 4 Or, cot A = Consider a right triangle ABC, right-angled at point B. 

If AB is 4k, then BC will be 3k, where k is a positive integer.

In ∆ABC,

(AC)2 = (AB)2 + (BC)2

= (4k)2 + (3k)2

= 16k2 + 9k2

= 25k2

AC = 5k 

10. In ∆PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

Ans: Given that, PR + QR = 25

 PQ = 5 

Let PR be x. 

Therefore, QR = 25 − x

Applying Pythagoras theorem in ∆PQR, we obtain

PR2 = PQ2 + QR2

x2 = (5)2 + (25 − x)2

x2 = 25 + 625 + x2 − 50x

50x = 650

x = 13

Therefore, PR = 13 cm

QR = (25 − 13) cm = 12 cm

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

(i) The value of tan A is always less than 1.

Ans: (i) Consider a ∆ABC, right-angled at B.

So, tan A < 1 is not always true.

Hence, the given statement is false.

Let AC be 12k, AB will be 5k, where k is a positive integer.

Applying Pythagoras theorem in ∆ABC, we obtain

AC2 = AB2 + BC2

(12k)2 = (5k)2 + BC2

144k2 = 25k2 + BC2

BC2 = 119k2.         

BC = 10.9k

It can be observed that for given two sides AC = 12k and AB = 5k,

BC should be such that,

AC − AB < BC < AC + AB

12k − 5k < BC < 12k + 5k

7k < BC < 17k

However, BC = 10.9k. Clearly, such a triangle is possible and hence, such value of sec A is possible.

Hence, the given statement is true. 

(iii) cos A is the abbreviation used for the cosecant of angle A.

Ans: Abbreviation used for cosecant of angle A is cosec A. And cos A is the abbreviation used for cosine of angle A.

Hence, the given statement is false. 

(iv) cot A is the product of cot and A

Ans: Cot A is not the product of cot and A. It is the cotangent of ∠A. Hence, the given statement is false.

(v) sin θ = 3/4, for some angle θ

Ans: sin θ = ¾   We know that in a right-angled triangle,

In a right-angled triangle, the hypotenuse is always greater than the remaining two sides.

Therefore, such a value of sin θ is not possible.

Hence, the given statement is false.

Exercise 8.2

1. Evaluate the following:

(i) sin60° cos30° + sin30° cos 60°

Ans: sin60° cos30° + sin30° cos 60° 

(ii) 2tan245° + cos230° − sin260°

Ans: 

Ans:  

Ans: 

Ans: 

2. Choose the correct option and justify your choice. 

(a) sin 60° 

(b) cos 60° 

(c) tan 60° 

(d) sin 30°

Ans: (a) sin 60° 

(a) tan 90° 

(b) 1 

(c) sin 45° 

(d) 0 

Ans: (d) 0 

(iii) sin2A = 2sinA is true when A =

(a) 0° 

(b) 30° 

(c) 45° 

(d) 60° 

Ans: (a) 0° 

(a) cos 60° 

(b) sin 60° 

(c) tan 60° 

(d) sin 30° 

Ans: (c) tan 60° 

3.

Ans: tan ( A + b) √3

tan (A + B)  = tan 60

⇒ A + B  = 60 ………………….(1)

⇒ A – B = 30……………………..(2) 

On adding both equations, we obtain

2A = 90

⇒ A = 45

From equation (1), we obtain

45 + B = 60

B = 15

Therefore, ∠A = 45° and ∠B = 15°

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

(i) sin (A + B) = sin A + sin B

Ans: sin (A + B) = sin A + sin B Let A = 30° and B = 60°

sin (A + B) = sin (30° + 60°)

= sin 90° = 1

And sin A + sin B = sin 30° + sin 60°

Clearly, sin (A + B) ≠ sin A + sin B

Hence, the given statement is false.

(ii) The value of sinθ increases as θ increases.

Ans: The value of sin θ increases as θ  increases in the interval of 0° < θ < 90° as sin 0° = 0 

(iii) The value of cos θ increases as θ increases

Ans: 

cos 90° = 0

It can be observed that the value of cos θ does not increase in the interval of 0°<θ<90°.

Hence, the given statement is false. 

(iv) sinθ = cos θ for all values of θ

Ans: sin θ = cos θ for all values of θ. This is true when θ = 45° 

It is not true for all other values of θ.

(v) cot A is not defined for A = 0° 

Ans: 

Exercises 8.3

1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Ans:

2. Write all the other trigonometric ratios of  A in terms of sec A.

Ans: We know that,

3. Choose the correct option. Justify your choice.

(i) 9 sec2  A – 9 tan2  A =

(a) 1

(b) 9

(c) 8

(d) 0

Ans: (b) 9

(ii) (1 + tan  + sec) (1 + cot  – cosec) =

(a) 0

(b) 1

(c) 2

(d) –1

Ans: (a) 0

(iii) (sec A + tan A) (1 – sin A) =

(a) sec A.

(b) sin A.

(c) cosec A.

(d) cos A.

Ans: (d) cos A.

(iv) 1 +| tan2 A / 1 + Cot2 A

Ans: Do yourself.

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

(i) 

Ans: 

Ans: 

Ans:  

Ans: 

Ans: 

Ans: 

(viii) sinA+cosecA)2 + (cos A + sec(A)= 7 + tan2 A + cot2 A

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