Question

Find the value of $\sin \theta$, if $\tan \theta =\frac{3}{4}$ and ${180}^{\circ }<\theta <{270}^{\circ }$.

• A. $\frac{4}{3}$
• B. $-\frac{4}{3}$
• C. $-\frac{3}{4}$
• D. $\frac{3}{5}$
• E. $-\frac{3}{5}$

Answer 1

The correct option is E $-\frac{3}{5}$

$1+{\tan }^2\theta ={\sec }^2\theta$

Hence ${\sec }^2\theta =1+\frac{9}{16}=\frac{16+9}{16}=\frac{25}{16}$.
Hence $\sec \theta =\frac{-5}{4}$ since $\theta$ is in $3^{rd}$ quadrant and in $3^{rd}$ quadrant only $\tan$ and $\cot$ are positive.
Therefore $\cos \theta =\frac{-4}{5}$.

Hence $\sin \theta =-\sqrt{1-{\cos }^2\theta }$

$=-\sqrt{1-\frac{16}{25}}$

$=-\sqrt{\frac{25-16}{25}}$

$=-\sqrt{\frac{9}{25}}$

$=\frac{-3}{5}$