Question

if 180°<theta<270, sin theta =-3/5, then cos theta/2=

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

Step-by-step explanation:

Given the trigonometric ratio,

$sin\theta =\dfrac{-3}{5}$    

Using the trigonometric relation,

$sin^{2} {\theta}+cos^2{\theta}=1$

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

                $=1+\dfrac{9}{25}=\dfrac{25-9}{25} =\dfrac{16}{25}$

$\implies cos{\theta}=\sqrt{\dfrac{16}{25} }=\pm\dfrac{4}{5}$

Given the condition, $180^{o} < \theta < 270 ^{o}$

The angle θ lies in 3rd quadrant. So sinθ as well as cosθ are negative.

Therefore, $cos{\theta}=-{4/}{5}$

To find the value of θ/2, consider the trigonometric relation,

$cos2\theta=2cos^2\theta-1$

For θ/2, we can write the above relation as

$cos\theta=2cos^2\dfrac{\theta}{2} -1$

$\implies -\dfrac{4}{5} =2cos^2\dfrac{\theta}{2} -1\implies 1-\dfrac{4}{5} =2cos^2\dfrac{\theta}{2}$

$\implies 2cos^2\dfrac{\theta}{2}=\dfrac{1}{5} \implies cos^2\dfrac{\theta}{2}=\dfrac{1}{5} \times \dfrac{1}{2} =\dfrac{1}{10}$

$\implies cos\dfrac{\theta}{2}=\pm\dfrac{1}{\sqrt{10} }$

For θ,  $180^{o} < \theta < 270 ^{o}$ then for θ/2, $90^{o} < \dfrac{\theta}{2} < 135 ^{o}$.

Therefore, the angle θ lies in 2nd quadrant, therefore cosθ is negative.

Therefore, the value of $cos\dfrac{\theta}{2}=\dfrac{1}{\sqrt{10} }$.