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°.
Solution:
We will use the basic definitions of trigonometric ratios to solve the given problem.
(i) sin (A + B) = sin A + sin B.
For the purpose of verification, Let A = 30° and B = 60°
L.H.S = sin (A + B)
= sin (30° + 60°)
= sin 90°
= 1
R.H.S = sin A + sin B
= sin 30° + sin 60°
= 1/2 + √3/2
= (1 + √3)/2
Since, sin (A + B) ≠ sin A + sin B.
Hence, the given statement is false.
(ii) The value of sin θ increases from 0 to 1 as θ increases from 0° to 90°
sin 0° = 0
sin 30° = 1/2 = 0.5
sin 45° = 1/√2 = 0.707
sin 60° = √3/2 = 0.866
sin 90° = 1
Hence, the given statement is true.
(iii) The value of cos θ decreases from 1 to 0 as θ increases from 0° to 90°
cos 0° = 1
cos 30° = √3/2 = 0.866
cos 45° = 1/√2 = 0.707
cos 60° = 1/2 = 0.5
cos 90° = 0
Hence, the given statement is false.
(iv) sin θ = cos θ for all values of θ, this is true when θ = 45°
As sin 45° = 1/√2 and cos 45° = 1/√2
It is not true for other values of θ
sin 30° = 1/√2 and cos 30° = √3/2
sin 60° = √3/2 and cos 60° = 1/√2
sin 90° = 1 and cos 90° = 0
Hence, the given statement is false.
(v) cot A = cos A/sin A
cot 0° = cos 0°/sin 0° = 1/0 = undefined
Hence, the given statement is true.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 8
Video Solution:
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°
Maths NCERT Solutions Class 10 Chapter 8 Exercise 8.2 Question 4
Summary:
The statements (i) sin (A + B) = sin A + sin B, (iii) The value of cos θ increases as θ increase, (iv) sinθ = cosθ for all values of θ are false and the statements (ii) The value of sin θ increases as θ increases , and (v) cot A is not defined for A = 0° are true.
☛ Related Questions:
- Evaluate the following: (i) sin 60° cos 30° + sin 30° cos 60° (ii) 2 tan² 45° + cos² 30° - sin² 60° (iii) cos 45°/(sec 30° + cosec 30°) (iv) sin 30° + tan 45° - cosec 60°/(sec 30° + cos 60° - cot 45°)
- Choose the correct option and justify your choice:(i) 2 tan 30°/1 + tan2 30°(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 60°(ii) 1 - tan2 45°/1 + tan2 45°(A) tan 90° (B) 1 (C) sin 45° (D) 0°(iii) sin 2A = 2 sin A is true when A =(A) 0° (B) 30°(C) 45° (D) 60°(iv) 2 tan 30°/1 - tan2 30°(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
- If tan (A + B) = √3 and tan (A - B) = 1/√3; 0° < (A + B) ≤ 900 , A > B, find A and B.
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