NCERT Solutions Class 10 Mathematics Chapter 8 Introduction To Trigonometry Download In Pdf

Chapter 8 Introduction To Trigonometry Download in pdf

EXERCISE 8.1

**Question 1.** In Δ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :

(i) sin A, cos A

(ii) sin C, cos C

**Question 2.** In Fig. 8.13, find tan P – cot R.

**Question 3.** If sin A =
3 ,
4
calculate cos A and tan A.

**Question 4. **Given 15 cot A = 8, find sin A and sec A.

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

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

**Question 7.** If cot θ =
7 ,
8
evaluate :

(i)
(1 sin ) (1 sin ) ,
(1 cos ) (1 cos )
+ θ − θ
+ θ − θ

(ii) cot2 θ

**Question 8.** If 3 cot A = 4, check whether
2
2
1 tan A
1 + tan A
−
= cos2 A – sin2A or not.

**Question 9.** In triangle ABC, right-angled at B, if tan A =
1 ,
3
find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C

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

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

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

(ii) sec A =
12
5 for some value of angle A.

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

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

(v) sin θ =
4
3 for some angle θ.

EXERCISE 8.2

**Question 1. **Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°

(ii) 2 tan2 45° + cos2 30° – sin2 60°

**Question 3.** If tan (A + B) = 3 and tan (A – B) =
1
3 ; 0° < A + B ≤ 90°; A > B, find A and B.

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

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

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

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

(iv) sin θ = cos θ for all values of θ.

(v) cot A is not defined for A = 0°.

EXERCISE 8.3

**Question 1.** Evaluate :

(i)
sin 18
cos 72
°
°

(ii)
tan 26
cot 64
°
°

(iii) cos 48° – sin 42°

(iv) cosec 31° – sec 59°

**Question 2.** Show that :

(i) tan 48° tan 23° tan 42° tan 67° = 1

(ii) cos 38° cos 52° – sin 38° sin 52° = 0

**Question 3.** If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.

**Question 4.** If tan A = cot B, prove that A + B = 90°.
5. If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.

EXERCISE 8.4

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

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

**Question 3.** Evaluate :

(i)
2 2
2 2
sin 63 sin 27
cos 17 cos 73
° + °
° + °

(ii) sin 25° cos 65° + cos 25° sin 65°

**Question 4.** Choose the correct option. Justify your choice.

(i) 9 sec2 A – 9 tan2 A =

(A) 1

(B) 9

(C) 8

(D) 0

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

(A) 0

(B) 1

(C) 2

(D) –1

(iii) (sec A + tan A) (1 – sin A) =

(A) sec A

(B) sin A

(C) cosec A

(D) cos A

(iv)
2
2
1 tan A
1 + cot A
+
=

(A) sec2 A

(B) –1

(C) cot2 A

(D) tan2 A

**Question 5.** Prove the following identities, where the angles involved are acute angles for which the
expressions are defined.

(i) (cosec θ – cot θ)2 =
1 cos
1 cos
− θ
+ θ

(ii) cos A 1 sin A 2 sec A

(iii)
tan cot 1 sec cosec
1 cot 1 tan
θ θ
+ = + θ θ
− θ − θ
[Hint : Write the expression in terms of sin θ and cos θ]

(iv)
1 sec A sin2 A
sec A 1 – cos A
+
= [Hint : Simplify LHS and RHS separately]

(v) cos A – sin A + 1 cosec A + cot A,
cos A + sin A – 1
= using the identity cosec2 A = 1 + cot2 A.

(vi)
1 sinA sec A + tan A
1 – sin A
+
=

(vii)
3
3
sin 2 sin tan
2 cos cos
θ − θ
= θ
θ − θ

(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A

(ix)
(cosec A – sin A)(sec A – cos A) 1
tanA + cot A
=
[Hint : Simplify LHS and RHS separately]

(x)
2 2
2
1 tan A 1 tanA
1 + cot A 1 – cot A
= tan2 A

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- Chapter 3 Pair of Linear Equations In Two Variables
- Chapter 4 Quadratic Equations
- Chapter 7 Coordinate Geometry
- Chapter 8 Introduction To Trigonometry
- Chapter 11 Constructions
- Chapter 9 Some Applications of Trigonometry
- chapter 10 Circles
- Chapter 12 Areas Related To Circles
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability
- Chapter 5 Arithmetic Progressions

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