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 $\mathrm{cosec}\theta$ 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 $\cot \theta =\frac{\cos \theta }{\sin \theta }=\frac{-4}{5}\times \frac{5}{3}=\frac{-4}{3}$
$\therefore (2\sec \theta -3\cot \theta )=\left\{2\times \frac{(-5)}{4}-3\times \frac{(-4)}{3}\right\}$
$=\left(\frac{-5}{2}+4\right)=\frac{3}{2}$