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# 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.