Answer
$\text{Answers in the following table:}$
$\begin{array}{cccccc}{\theta} & {-\pi} & {-2 \pi / 3} & {0} & {\pi / 2} & {3 \pi / 4} \\ \hline \sin \theta & {0} & {-\frac{\sqrt{3}}{2}} & {0} & {1} & {\frac{1}{\sqrt{2}}} \\ {\cos \theta} & {-1} & {-\frac{1}{2}} & {1} & {0} & {-\frac{1}{\sqrt{2}}} \\ {\tan \theta} & {0} & {\sqrt{3}} & {0} & {\mathrm{UND}} & {-1} \\ {\cot \theta} & {\mathrm{UND}} & {\frac{1}{\sqrt{3}}} & {\mathrm{UND}} & {0} & {-1} \\ {\sec \theta} & {-1} & {-2} & {1} & {\mathrm{UND}} & {-\sqrt{2}} \\ {\sec \theta} & {\mathrm{UND}} & {-\frac{2}{\sqrt{3}}} & {\mathrm{UND}} & {1} & {\sqrt{2}}\end{array}$
Work Step by Step
$\text{Answer in the following table:}$
$\begin{array}{cccccc}{\theta} & {-\pi} & {-2 \pi / 3} & {0} & {\pi / 2} & {3 \pi / 4} \\ \hline \sin \theta & {0} & {-\frac{\sqrt{3}}{2}} & {0} & {1} & {\frac{1}{\sqrt{2}}} \\ {\cos \theta} & {-1} & {-\frac{1}{2}} & {1} & {0} & {-\frac{1}{\sqrt{2}}} \\ {\tan \theta} & {0} & {\sqrt{3}} & {0} & {\mathrm{UND}} & {-1} \\ {\cot \theta} & {\mathrm{UND}} & {\frac{1}{\sqrt{3}}} & {\mathrm{UND}} & {0} & {-1} \\ {\sec \theta} & {-1} & {-2} & {1} & {\mathrm{UND}} & {-\sqrt{2}} \\ {\sec \theta} & {\mathrm{UND}} & {-\frac{2}{\sqrt{3}}} & {\mathrm{UND}} & {1} & {\sqrt{2}}\end{array}$
We can find these values without using a calculator by considering the standard unit circle values and remembering that $\sin\theta=y$ and $\cos\theta=x$ along the unit circle of radius 1.
