Question

If θ is an angle in standard position and its terminal side passes through the point (1,-8), find the exact value of csc θ in simplest radical form.

Answer 1

Check the picture below.

well, we know the angle's cosine and sine or namely adjacent and opposite sides, let's get the hypotenuse.

`\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2 \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{1}\\ b=\stackrel{opposite}{-8}\\ \end{cases} c=√(1^2+(-8)^2)\implies c=√(65) \\\\[-0.35em] ~\dotfill\\\\ csc(\theta )=\cfrac{\stackrel{hypotenuse}{√(65)}}{\underset{opposite}{-8}}\implies csc(\theta )=-\cfrac{√(65)}{8}`