Question

If $\theta$ lies in second quadrant and $\sin \theta=\frac{3}{5}$, then which of the following is true ?

• A. $\sec \theta=-\frac{5}{4}$
• B. $\cot \theta=-\frac{4}{3}$
• C. $2 \sec \theta=3 \cot \theta=\frac{3}{2}$
• D. All the above are true

Answer 1

Answer: (D)

In II quadrant, only $\sin \theta$ and $\operatorname{cosec} \theta$ are positive.
$\therefore \cos ^2 \theta =\left(1-\sin ^2 \theta\right)=\left(1-\frac{9}{25}\right)=\frac{16}{25} $
$\Rightarrow \cos \theta =-\sqrt{\frac{16}{25}}=\frac{-4}{5} $
$\therefore \sec \theta =\frac{-5}{4} $
and $ \cot \theta =\frac{\cos \theta}{\sin \theta}=\frac{-4}{5} \times \frac{5}{3}=\frac{-4}{3} $
$\therefore (2 \sec \theta-3 \cot \theta) =\left\{2 \times \frac{(-5)}{4}-3 \times \frac{(-4)}{3}\right\} $
$=\left(\frac{-5}{2}+4\right)=\frac{3}{2}$