Class 10 Maths (NCERT)
Chapter 8: Introduction to Trigonometry
Exercise 8.2 – Stepwise Solutions
Standard Values Used
sin 30° = 1/2 cos 30° = √3/2 tan 30° = 1/√3
sin 45° = √2/2 cos 45° = √2/2 tan 45° = 1
sin 60° = √3/2 cos 60° = 1/2 tan 60° = √3
1. Evaluate the following:
(i) sin 60° cos 30° + sin 30° cos 60°
= (√3/2 × √3/2) + (1/2 × 1/2)
= (3/4) + (1/4)
= 1
Answer = 1
(ii) 2 tan²45° + cos²30° − sin²60°
= 2(1)² + (√3/2)² − (√3/2)²
= 2 + 3/4 − 3/4
= 2
Answer = 2
(iii) cos 45° / (sec 30° + cosec 30°)
sec 30° = 2/√3
cosec 30° = 2
Denominator = 2/√3 + 2
Numerator = √2/2
After simplification:
Answer = √2 / (2 + 2√3)
(iv) (sin 30° + tan 45° − cosec 60°) / (sec 30° + cos 60° + cot 45°)
Numerator = 1/2 + 1 − 2/√3
Denominator = 2/√3 + 1/2 + 1
After simplification:
Answer = 0
(v) (5 cos²60° + 4 sec²30° − tan²45°) / (sin²30° + cos²30°)
cos²60° = (1/2)² = 1/4
sec²30° = (2/√3)² = 4/3
tan²45° = 1
Numerator = 5(1/4) + 4(4/3) − 1
= 5/4 + 16/3 − 1
Denominator = (1/2)² + (√3/2)²
= 1/4 + 3/4 = 1
After simplifying numerator:
= 67/12
Answer = 67/12
2. Choose the correct option
(i) 2 tan 30° / (1 + tan²30°)
tan 30° = 1/√3
= 2(1/√3) / (1 + 1/3)
= (2/√3) / (4/3)
= (2/√3 × 3/4)
= 1/2
1/2 = sin 30°
Correct option: (D) sin 30°
(ii) (1 − tan²45°) / (1 + tan²45°)
= (1 − 1) / (1 + 1)
= 0/2 = 0
Correct option: (D) 0
(iii) sin 2A = 2 sin A is true when A = ?
sin 2A = 2 sin A cos A
So 2 sin A cos A = 2 sin A
Divide both sides by 2 sin A:
cos A = 1
This happens when A = 0°
Correct option: (A) 0°
(iv) 2 tan 30° / (1 − tan²30°)
= (2/√3) / (1 − 1/3)
= (2/√3) / (2/3)
= (2/√3 × 3/2)
= √3
√3 = tan 60°
Correct option: (C) tan 60°
3. If tan(A+B) = √3 and tan(A−B) = 1/√3
tan(A+B) = √3 ⇒ A+B = 60°
tan(A−B) = 1/√3 ⇒ A−B = 30°
Add equations:
2A = 90° ⇒ A = 45°
Substitute:
45° + B = 60° ⇒ B = 15°
Answer: A = 45°, B = 15°
4. State whether true or false
(i) sin(A+B) = sin A + sin B
False (Formula is sinA cosB + cosA sinB)
(ii) The value of sin θ increases as θ increases
True (for 0° ≤ θ ≤ 90°)
(iii) The value of cos θ increases as θ increases
False (cos decreases as angle increases)
(iv) sin θ = cos θ for all θ
False (only true when θ = 45°)
(v) cot A is not defined for A = 0°
cot 0° = cos0°/sin0° = 1/0 (undefined)
True
Final Answers (Quick)
1(i) 1
1(ii) 2
1(iii) √2/(2+2√3)
1(iv) 0
1(v) 67/12
2(i) sin30°
2(ii) 0
2(iii) 0°
2(iv) tan60°
3) A=45°, B=15°
4) F, T, F, F, T