Finding the Trigonometric Functions of an Angle

Find $\sin t, \cos t, \tan t, \sec t, \csc t,$ and $\cot t .$ when $t = \frac{π}{6} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We have previously used the properties of equilateral triangles to demonstrate that $\sin \frac{π}{6} = \frac{1}{2}$ and $\cos \frac{π}{6} = \frac{\sqrt{3}}{2} .$ We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

$\begin{matrix} tan \frac{π}{6} & = \frac{sin \frac{π}{6}}{cos \frac{π}{6}} \\ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} & = \frac{1}{\sqrt{3}} & = \frac{\sqrt{3}}{3} \\ sec \frac{π}{6} & = \frac{1}{cos \frac{π}{6}} \\ = \frac{1}{\frac{\sqrt{3}}{2}} & = \frac{2}{\sqrt{3}} & = \frac{2 \sqrt{3}}{3} \\ \csc \frac{π}{6} & = \frac{1}{\sin \frac{π}{6}} & = \frac{1}{\frac{1}{2}} & = 2 \\ cot \frac{π}{6} & = \frac{cos \frac{π}{6}}{sin \frac{π}{6}} \\ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} & = \sqrt{3} \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find $\sin t, \cos t, \tan t, \sec t, \csc t,$ and $\cot t .$ when $t = \frac{π}{3} .$

How did it go?