Question

If ${180}^{\circ }<\theta <{270}^{\circ },sin\theta =-3/5,cot\theta =4/3$, then $\cos \left(\frac{\theta }{2}\right)=$

• A. $-1/\sqrt{10}$
• B. $1/\sqrt{10}$
• C. $-1/10$
• D. 10

Answer 1

Answer: (A)

$-1/\sqrt{10}$

${180}^{\circ }\theta <{270}^{\circ }\Rightarrow {90}^{\circ }<\frac{\theta }{2}<{135}^{\circ }$
${\cos }^2\theta =1-{\sin }^2\theta =1-\frac{9}{25}=\frac{16}{25}$

${180}^{\circ }<\theta <{270}^{\circ }$

${\cos }^2\theta =\frac{16}{25}\Rightarrow \cos \theta =-\frac{4}{5}$

${\cos }^2\frac{\theta }{2}=\frac{1+\cos \theta }{2}=\frac{1-4/5}{2}=\frac{1}{10}$

${90}^{\circ }<\theta /2<{135}^{\circ }$
${\cos }^2\left(\theta /2\right)=\frac{1}{10}$
$\Rightarrow \cos \left(\theta /2\right)=-\frac{1}{\sqrt{10}}$