Question

State whether the following are true or false. Justify your answer.
(i) $\sin (A+B)=\sin A+\sin B$
(ii) The value of $\sin \theta$ increases as $\theta$ increases.
(iii) The value of $\cos \theta$ increases as increases.
(iv) $\sin \theta =\cos \theta$ for all values of $\theta$.
(v) $\cot A$ is not defined for $A=0^{\circ }$.

Answer 1

(i) False

$\sin (A+B)=\sin A+\sin B$

$A={45}^o,B={45}^o$

LHS: $\sin ({45}^o+{45}^o)=\sin {90}^o=1$

RHS : $\sin A+\sin B=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$

$LHS\ne RHS$

(ii) True

$\sin 0^o=0\sin {30}^o=\frac{1}{2}=\sin {45}^o=\frac{1}{\sqrt{2}},\sin {60}^o=\frac{\sqrt{3}}{2}$

$\sin {90}^o=1$

$\therefore \theta$ increase ans value of $\sin \theta$ also increase.

(iii) False

$\cos 0=1,\cos 30=\frac{13}{2},\cos 45=\frac{1}{\sqrt{2}}$

$\cos 60=\frac{1}{2},\cos 90=0$

Value of $\cos \theta$ does not increase from $0<0<{90}^o$

(iv) False

This is true for $\theta ={45}^o$

Let $\theta ={90}^o$

$\sin 90=1,\cos 90=0$

(v) True

$\cot A=\frac{\cos A}{\sin A}$

$\cot 0^o=\frac{\cos 0}{\sin 0}=\frac{1}{0}=$not defined