5.3 The other trigonometric functions  (Page 4/13)

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.

1. $$\begin{array}{l}\tan (\mathrm{45°})=\frac{\sin (\mathrm{45°})}{\cos (\mathrm{45°})}\hfill \\ =\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\hfill \\ =1\hfill \end{array}$$
2. $$\begin{array}{l}\sec \left(\frac{5\pi }{6}\right)=\frac{1}{\cos \left(\frac{5\pi }{6}\right)}\hfill \\ =\frac{1}{-\frac{\sqrt{3}}{2}}\hfill \\ \begin{array}{l}=\frac{-2\sqrt{3}}{1}\hfill \\ =\frac{-2}{\sqrt{3}}\hfill \\ =-\frac{2\sqrt{3}}{3}\hfill \end{array}\hfill \end{array}$$

We can simplify this by rewriting both functions in terms of sine and cosine.

By showing that $\frac{\sec t}{\tan t}$ can be simplified to $\csc t,$ we have, in fact, established a new identity.

We can find the sine using the Pythagorean Identity, ${\cos }^{2}t+{\sin }^{2}t=1,$ and the remaining functions by relating them to sine and cosine.

The sign of the sine depends on the y -values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y -values are negative, its sine is negative, $-\frac{5}{13}.$

The remaining functions can be calculated using identities relating them to sine and cosine.