Question

question 3
1 pts
if ( 180^{circ}<\theta<270^{circ} ) and ( cos \theta=-\frac{7}{15} ), find ( cos \frac{\theta}{2} ).
( \frac{2 sqrt{15}}{15} )
( -\frac{sqrt{15}}{15} )
( -\frac{2 sqrt{15}}{15} )
( \frac{sqrt{15}}{15} )

Explanation:

Step1: Determine quadrant of $\theta/2$

Given $180^\circ < \theta < 270^\circ$, divide by 2: $90^\circ < \theta/2 < 135^\circ$ (second quadrant, cosine negative).

Step2: Use half-angle formula

Half-angle formula: $\cos(\theta/2) = -\sqrt{\frac{1+\cos\theta}{2}}$ (negative sign for second quadrant).

Step3: Substitute $\cos\theta = -7/15$

$\cos(\theta/2) = -\sqrt{\frac{1 - 7/15}{2}} = -\sqrt{\frac{8/15}{2}} = -\sqrt{\frac{4}{15}} = -\frac{2}{\sqrt{15}} = -\frac{2\sqrt{15}}{15}$

Answer:

$-\frac{2\sqrt{15}}{15}$