Answer
$\sin\theta$ = $ - \frac{7}{25}$
$\tan\theta$ = $ - \frac{7}{24}$
$\cot\theta$ = $ - \frac{24}{7}$
$\sec\theta$ =$ \frac{25}{24}$
$\csc\theta$ = $ - \frac{25}{7}$
Work Step by Step
Given-
$\cos\theta$ = $\frac{24}{25}$ and $\theta$ terminates in QIV
Therefore $\frac{x}{r}$ = $\frac{24}{25}$
We may consider $x = 24$ and $r = 25$
As $x^{2} + y^{2}$ = $r^{2}$
Therefore $y$ = $\sqrt {r^{2} - x^{2}}$ = $\sqrt {25^{2} - 24^{2}}$
Or $y$ = $\sqrt {625 - 576}$ = $\sqrt {49}$ = $± 7$
As $\theta$ terminates in QIV, y will be negative
Therefore, $y$ = $-7$
i.e $x = 24$, $y$ = $-7$ and $r = 25$
Now we can write all T-functions of $\theta$ using Definition-I as following-
$\sin\theta$ =$ \frac{y}{r}$ = $ \frac{-7}{25}$ = $ - \frac{7}{25}$
$\tan\theta$ =$ \frac{y}{x}$ =$ \frac{-7}{24}$ = $ - \frac{7}{24}$
$\cot\theta$ =$ \frac{x}{y}$ =$ \frac{24}{-7}$ = $ - \frac{24}{7}$
$\sec\theta$ =$ \frac{r}{x}$ =$ \frac{25}{24}$
$\csc\theta$ =$ \frac{r}{y}$ =$ \frac{25}{-7}$ = $ - \frac{25}{7}$
