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 sinθ increases as θ increases
(iii) The value of cos θ increases as θ increases
(iv) sinθ = cos θ for all values of θ
(v) cot A is not defined for A = 0°
Answer:
(i) sin (A + B) = sin A + sin B
Let A = 30° and B = 60°
sin (A + B) = sin (30° + 60°)
= sin 90°
= 1
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 sin θ increases as θ increases in the interval of 0° < θ < 90° as
sin 0° = 0
sin 30º = 1/2 = 0.5
sin 45º = 1/√2 = 0.707
sin 60° = √3/2 = 0.866
sin 90° = 1
Hence, the given statement is true.
(iii) cos 0° = 1
cos 30° = √3/2 = 0.866
cos 45° = 1/√2 = 0.707
cos 60° = 1/2 = 0.5
cos90° = 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) sin θ = cos θ for all values of θ.
This is true when θ = 45°
As sin 45° = 1/√2
cos 45° = 1/√2
It is not true for all other values of θ.
As sin 30° = 1/2 and cos 30° = √3/2,
Hence, the given statement is false.
(v) cot A is not defined for A = 0°
As cot A = cos A/sin A,
cot 0° = cos 0°/sin 0° = 1/0 = undefined
Hence, the given statement is true.
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