State whether the following is true or false. Justify your answer.
The value of $\sin \theta$ increases as $\theta$ increases.

State whether the following are true or false. Justify your answer.
$(i)$ The value of $\tan A$ is always less than $1$.
$(ii)$ $\sec A = \frac{12}{5}$ for some value of angle $A$.

State whether the following is true or false. Justify your answer.
$\sin (A+B) = \sin A + \sin B$

Given $\tan A = \frac{4}{3},$ find the other trigonometric ratios of the $\angle A$.

Consider $\triangle ACB$,right-angled at $C$,in which $AB = 29$ units,$BC = 21$ units and $\angle ABC = \theta$. Determine the values of:
$(i)$ $\cos^2 \theta + \sin^2 \theta$
$(ii)$ $\cos^2 \theta - \sin^2 \theta$

In $\triangle ABC$,right-angled at $B$,$AB = 24 \, cm$,$BC = 7 \, cm$. Determine:
$(i)$ $\sin A, \cos A$
$(ii)$ $\sin C, \cos C$