Question

If 180 degrees 270degrees and sin theta=-4/5 calculate sin (theta/2)and cos(theta/2)

Answer 1

Answer:

$\sin( \alpha ) = - \frac{4}{5} \: \: then \: \: \cos( \alpha ) = - \frac{3}{5}$

given

$180 < \alpha < 270 \: \: then \: \: 90 < \frac{ \alpha }{2} < 135$

now,

$2 \cos {}^{2} ( \frac{ \alpha }{2} ) - 1 = \cos( \alpha ) = - \frac{3}{5}$

$\cos {}^{2} ( \frac{ \alpha }{2} ) = (1 - \frac{3}{5} ) \div 2 = \frac{1}{5}$

$\cos( \frac{ \alpha }{2} ) = - \frac{1}{ \sqrt{5} } \: \: and \: \: \sin( \frac{ \alpha }{2} ) = \frac{2}{ \sqrt{5} }$

in 2nd quadrant, Cos is -ve & Sin is + ve.