Q4. State whether the following are true of 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 $$\theta$$ increases.

(iv) $$sin\theta \ = \ cos\theta$$ for all values of $$\theta$$.

(v) cot A is not defined for A = 0°.

(i) False, because
for a case when A = 60° and B = 30°,

$$sin (A + B) \ = \ sin (60° + 30°) \ = \ sin 90° \ = \ 1$$

and, $$sin A \ + \ sin B \ = \ sin 60° \ + \ sin 30° \ = \frac{ \sqrt{3}}{2} \ + \ \frac{1}{2} \ = \ \frac{ \sqrt{3} + 1}{2}$$

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

(ii) True, as we can see from the table as the value of $$sin \theta$$ increases $$\theta$$ also increases.

(iii) False, as we can see from the table as the value of $$cos\theta$$ increases $$\theta$$ decreases.

(iv) False, as it is only true for $$\theta \ = \ 45°$$.

(v) True, as $$tan 0° \ = \ 0$$ and $$cot0° \ = \ \frac{1}{tan0°} \ = \ \frac{1}{0}$$ ,i.e., not defined.