Question

Find sin θ/2 when 270º<θ<360º and cosθ is defined as:

cosθ = √7/4

Answer 1

$\bf sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1-cos({{ \theta}})}{2}}\\\\ -------------------------------\\\\ sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1-\frac{\sqrt{7}}{4}}{2}}\implies sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{\frac{4-\sqrt{7}}{4}}{2}} \\\\\\ sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{\frac{4-\sqrt{7}}{4}}{\frac{2}{1}}}\implies sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{4-\sqrt{7}}{4}\cdot \cfrac{1}{2}}$

$\bf sin\left( \cfrac{\theta }{2} \right)=\pm\sqrt{\cfrac{4-\sqrt{7}}{8}}\implies sin\left( \cfrac{\theta }{2} \right)=\pm\sqrt{\cfrac{1}{2}-\cfrac{\sqrt{7}}{8}} \\\\\\ sin\left( \cfrac{\theta }{2} \right)=-\sqrt{\cfrac{1}{2}-\cfrac{\sqrt{7}}{8}}\impliedby \textit{on the \underline{IV quadrant}}$