7.4 The other trigonometric functions  (Page 5/14)

Finding the values of trigonometric functions

Find the values of the six trigonometric functions of angle $t$ based on [link] .

$$\begin{array}{cccc}\hfill \text{sin }t& =y\hfill & =-\frac{\sqrt{3}}{2}\hfill & \\ \hfill \text{cos }t& =x\hfill & =-\frac{1}{2}\hfill & \\ \hfill \text{tan }t& =\frac{\text{sin }t}{\text{cos }t}\hfill & =\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\hfill & =\sqrt{3}\hfill \\ \hfill \text{sec }t& =\frac{1}{\text{cos }t}\hfill & =\frac{1}{-\frac{1}{2}}\hfill & =\mathrm{-2}\hfill \\ \hfill \text{csc }t& =\frac{1}{\text{sin }t}\hfill & =\frac{1}{-\frac{\sqrt{3}}{2}}\hfill & =-\frac{2\sqrt{3}}{3}\hfill \\ \hfill \text{cot }t& =\frac{1}{\text{tan }t}\hfill & =\frac{1}{\sqrt{3}}\hfill & =\frac{\sqrt{3}}{3}\hfill \end{array}$$

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Try It

Find the values of the six trigonometric functions of angle $t$ based on [link] .

$\begin{array}{l}\sin t=-1,\cos t=0,\tan t=\text{Undefined}\\ \sec t=\text{Undefined,}\csc t=-1,\cot t=0\end{array}$

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Evaluating trigonometric functions with a calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor $\frac{\pi }{180}$ to convert the degrees to radians. To find the secant of $\mathrm{30°},$ we could press