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NCERT Solutions · Maths Ch 8
Introduction to Trigonometry
49 questions · 3 exercises
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Ex 8.1
Ex 8.2
Ex 8.3
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
18 QUESTIONS
Exercise 8.1
Q1(i)
In △ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: sin A, cos A
(2 marks)
Q1(ii)
In △ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: sin C, cos C
(1 mark)
Q2
In Fig. 8.13, find tan P – cot R.
Figure: △PQR right-angled at Q, with PQ = 12 cm and PR = 13 cm (the hypotenuse).
(2 marks)
Q3
If sin A = 3/4, calculate cos A and tan A.
(2 marks)
Q4
Given 15 cot A = 8, find sin A and sec A.
(2 marks)
Q5
Given sec θ = 13/12, calculate all other trigonometric ratios.
(3 marks)
Q6
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
(2 marks)
Q7(i)
If cot θ = 7/8, evaluate: (1 + sin θ)(1 – sin θ) / [(1 + cos θ)(1 – cos θ)]
(2 marks)
Q7(ii)
If cot θ = 7/8, evaluate: cot²θ
(1 mark)
Q8
If 3 cot A = 4, check whether (1 – tan²A)/(1 + tan²A) = cos²A – sin²A or not.
(3 marks)
Q9(i)
In triangle ABC, right-angled at B, if tan A = 1/√3, find the value of: sin A cos C + cos A sin C
(2 marks)
Q9(ii)
In triangle ABC, right-angled at B, if tan A = 1/√3, find the value of: cos A cos C – sin A sin C
(1 mark)
Q10
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.
(3 marks)
Q11(i)
State whether the following is true or false. Justify your answer: The value of tan A is always less than 1.
(1 mark)
Q11(ii)
State whether the following is true or false. Justify your answer: sec A = 12/5 for some value of angle A.
(1 mark)
Q11(iii)
State whether the following is true or false. Justify your answer: cos A is the abbreviation used for the cosecant of angle A.
(1 mark)
Q11(iv)
State whether the following is true or false. Justify your answer: cot A is the product of cot and A.
(1 mark)
Q11(v)
State whether the following is true or false. Justify your answer: sin θ = 4/3 for some angle θ.
(1 mark)
2
15 QUESTIONS
Exercise 8.2
Q1(i)
Evaluate: sin 60° cos 30° + sin 30° cos 60°
(2 marks)
Q1(ii)
Evaluate: 2 tan²45° + cos²30° – sin²60°
(2 marks)
Q1(iii)
Evaluate: cos 45° / (sec 30° + cosec 30°)
(3 marks)
Q1(iv)
Evaluate: (sin 30° + tan 45° – cosec 60°) / (sec 30° + cos 60° + cot 45°)
(3 marks)
Q1(v)
Evaluate: (5 cos²60° + 4 sec²30° – tan²45°) / (sin²30° + cos²30°)
(2 marks)
Q2(i)
Choose the correct option and justify your choice: 2 tan 30° / (1 + tan²30°) = (A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°
(1 mark)
Q2(ii)
Choose the correct option and justify your choice: (1 – tan²45°) / (1 + tan²45°) = (A) tan 90° (B) 1 (C) sin 45° (D) 0
(1 mark)
Q2(iii)
Choose the correct option and justify your choice: sin 2A = 2 sin A is true when A = (A) 0° (B) 30° (C) 45° (D) 60°
(1 mark)
Q2(iv)
Choose the correct option and justify your choice: 2 tan 30° / (1 – tan²30°) = (A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
(1 mark)
Q3
If tan (A + B) = √3 and tan (A – B) = 1/√3; 0° B, find A and B.
(3 marks)
Q4(i)
State whether the following is true or false. Justify your answer: sin (A + B) = sin A + sin B.
(1 mark)
Q4(ii)
State whether the following is true or false. Justify your answer: The value of sin θ increases as θ increases.
(1 mark)
Q4(iii)
State whether the following is true or false. Justify your answer: The value of cos θ increases as θ increases.
(1 mark)
Q4(iv)
State whether the following is true or false. Justify your answer: sin θ = cos θ for all values of θ.
(1 mark)
Q4(v)
State whether the following is true or false. Justify your answer: cot A is not defined for A = 0°.
(1 mark)
3
16 QUESTIONS
Exercise 8.3
Q1
Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
(3 marks)
Q2
Write all the other trigonometric ratios of ∠A in terms of sec A.
(3 marks)
Q3(i)
Choose the correct option. Justify your choice: 9 sec²A – 9 tan²A = (A) 1 (B) 9 (C) 8 (D) 0
(1 mark)
Q3(ii)
Choose the correct option. Justify your choice: (1 + tan θ + sec θ)(1 + cot θ – cosec θ) = (A) 0 (B) 1 (C) 2 (D) –1
(1 mark)
Q3(iii)
Choose the correct option. Justify your choice: (sec A + tan A)(1 – sin A) = (A) sec A (B) sin A (C) cosec A (D) cos A
(1 mark)
Q3(iv)
Choose the correct option. Justify your choice: (1 + tan²A)/(1 + cot²A) = (A) sec²A (B) –1 (C) cot²A (D) tan²A
(1 mark)
Q4(i)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (cosec θ – cot θ)² = (1 – cos θ)/(1 + cos θ)
(3 marks)
Q4(ii)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A
(3 marks)
Q4(iii)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: tan θ/(1 – cot θ) + cot θ/(1 – tan θ) = 1 + sec θ cosec θ [Hint: write the expression in terms of sin θ and cos θ]
(3 marks)
Q4(iv)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (1 + sec A)/sec A = sin²A/(1 – cos A) [Hint: simplify LHS and RHS separately]
(3 marks)
Q4(v)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (cos A – sin A + 1)/(cos A + sin A – 1) = cosec A + cot A, using the identity cosec²A = 1 + cot²A
(3 marks)
Q4(vi)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: √[(1 + sin A)/(1 – sin A)] = sec A + tan A
(3 marks)
Q4(vii)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (sin θ – 2 sin³θ)/(2 cos³θ – cos θ) = tan θ
(3 marks)
Q4(viii)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A
(3 marks)
Q4(ix)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A) [Hint: simplify LHS and RHS separately]
(3 marks)
Q4(x)
Prove the following identity, where the angles involved are acute angles for which the expressions are defined: (1 + tan²A)/(1 + cot²A) = [(1 – tan A)/(1 – cot A)]² = tan²A
(3 marks)