Horizontal alignment test

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Finding the trigonometric functions of an angle

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

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{array}{l} \tan \frac{π}{6} & = & \frac{\sin}{} \\ = & \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \end{array}

\begin{array}{l} \begin{array}{l} \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} \end{array} \end{array}

\csc \frac{π}{6} = \frac{1}{\sin \frac{π}{6}} = \frac{1}{\frac{1}{2}} = 2

\begin{array}{l} \begin{array}{l} \cot \frac{π}{6} = \frac{\cos \frac{π}{6}}{\sin \frac{π}{6}} \\ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \end{array} \end{array}

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Source: OpenStax, Horizontal alignment test. OpenStax CNX. Feb 24, 2015 Download for free at http://legacy.cnx.org/content/col11763/1.2

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