State whether the following statements 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 $A = 0^o$.

To find: 

We have to state whether the given statements are true or false.

Solution:  

(i) Let $\mathrm{A}=30^{\circ}$ and $\mathrm{B}=60^{\circ}$

This implies,

LHS $=\sin (30^{\circ}+60^{\circ})=\sin 90^{\circ}$

$=1$

RHS $=\sin 30^{\circ}+\sin 60^{\circ}$

$=\frac{1}{2}+\frac{\sqrt3}{2}$

$=\frac{1+\sqrt3}{2}$

LHS $≠$ RHS

Hence, the given statement is false.

(ii) We know that,

$\sin 0^{\circ}=0$

$\sin 30^{\circ}=\frac{1}{2}$

$\sin 45^{\circ}=\frac{1}{\sqrt{2}}$

$\sin 60^{\circ}=\frac{\sqrt{3}}{2}$

$\sin 90^{\circ}=1$

Therefore, the value of $\sin\ \theta$ increases as $\theta$ increases.

Hence, the given statement is true.

(iii) We know that,

$\cos 0^{\circ}=1$

$\cos 30^{\circ}=\frac{\sqrt3}{2}$

$\cos 45^{\circ}=\frac{1}{\sqrt{2}}$

$\cos 60^{\circ}=\frac{1}{2}$

$\cos 90^{\circ}=0$

Therefore, the value of $\cos\ \theta$ decreases as $\theta$ increases.

Hence, the given statement is false.

(iv) We know that,

$\sin 0^{\circ}=0$

$\cos 0^{\circ}=1$

$\sin 0^{\circ}≠\cos 0^{\circ}$

Therefore, $\sin \theta$ is not always equal to $\cos \theta$ 

Hence, the given statement is false.

(v) We know that,

$\cot 0^{\circ}=\frac{\cos 0^{\circ}}{\sin 0^{\circ}}$

$=\frac{1}{0}$

$\frac{1}{0}$ is not defined.

Hence, the given statement is true. 

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Updated on: 10-Oct-2022

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