Question

Find the exact values of the six trigonometric functions of the real number t

Answer 1

In a unit circle, given the (x,y) coordinate, x corresponds to cosine, and y corresponds to sine.

Then use the trigonometric identity to solve for tangent.

We therefore have the following ratios for sin, cos, and tan.

`\begin{gathered} \sin t=(15)/(17) \\ \cos t=-(8)/(17) \\ \tan t=(\sin t)/(\cos t)=((15)/(17))/(-(8)/(17))=-(15)/(8) \\ \\ \text{Therefore,} \\ \sin t=(15)/(17) \\ \cos t=-(8)/(17) \\ \tan t=-(15)/(8) \end{gathered}`

Solving for the reciprocal of sin, cos, and tan we have

`\begin{gathered} \csc t=\Big(\sin t\Big)^(-1)=\Big((15)/(17)\Big)^(-1)=(17)/(15) \\ \sec t=\Big(\cos t\Big)^(-1)=\Big(-(8)/(17)\Big)^(-1)=-(17)/(8) \\ \cot t=\Big(\tan t\Big)^(-1)=\Big(-(15)/(8)\Big)^(-1)=-(8)/(15) \\ \\ \text{Therefore,} \\ \csc t=(17)/(15) \\ \sec t=-(17)/(8) \\ \cot t=-(8)/(15) \end{gathered}`