Question

Need to get the values of each trig function using half angle formulas within the constraint

Answer 1

$\bf cot(u)=4\qquad \pi \ \textless \ u\ \textless \ \frac{3\pi }{2}\implies cot=\cfrac{-4}{-1}\cfrac{\leftarrow adjacent}{\leftarrow opposite} \\\\\\ hypotenuse\implies \sqrt{(-4)^2+(-1)^2}\implies \sqrt{17} \\\\\\ cos(\theta )=\cfrac{-4}{\sqrt{17}}\implies \cfrac{-4\sqrt{17}}{17}\\\\ -------------------------------$

$\bf sin\left( \frac{u}{2} \right)=\pm\sqrt{\cfrac{1+\frac{4\sqrt{17}}{17}}{2}}\implies \pm\sqrt{\cfrac{\frac{17+4\sqrt{17}}{17}}{2}}\implies \boxed{\pm\sqrt{\cfrac{17+4\sqrt{17}}{34}}} \\\\\\ cos\left( \frac{u}{2} \right)=\pm\sqrt{\cfrac{1-\frac{4\sqrt{17}}{17}}{2}}\implies \pm\sqrt{\cfrac{\frac{17-4\sqrt{17}}{17}}{2}}\implies \boxed{\pm\sqrt{\cfrac{17-4\sqrt{17}}{34}}}$

$\bf tan\left( \frac{u}{2} \right)=\pm\sqrt{\cfrac{\frac{17+4\sqrt{17}}{34}}{\frac{17-4\sqrt{17}}{34}}}\implies \boxed{\pm\sqrt{\cfrac{17+4\sqrt{17}}{17-4\sqrt{17}}}}$