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When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in [link] . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.
Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.
Using the triangle shown in [link] , evaluate $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha ,\mathrm{cos}\text{\hspace{0.17em}}\alpha ,\mathrm{tan}\text{\hspace{0.17em}}\alpha ,\mathrm{sec}\text{\hspace{0.17em}}\alpha ,\mathrm{csc}\text{\hspace{0.17em}}\alpha ,\text{and}\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\alpha .$
Using the triangle shown in [link] ,evaluate $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t,\text{cos}\text{\hspace{0.17em}}t,\text{tan}\text{\hspace{0.17em}}t,\text{sec}\text{\hspace{0.17em}}t,\text{csc}\text{\hspace{0.17em}}t,\text{and}\text{\hspace{0.17em}}\text{cot}\text{\hspace{0.17em}}t.$
$$\begin{array}{ccccccccc}\hfill \text{sin}t& =& \frac{33}{65},\hfill & \hfill \text{cos}t& =& \frac{56}{65},\hfill & \hfill \text{tan}t& =& \frac{33}{56},\hfill \\ \hfill \text{sec}t& =& \frac{65}{56},\hfill & \hfill \text{csc}t& =& \frac{65}{33},\hfill & \hfill \text{cot}t& =& \frac{56}{33}\hfill \end{array}$$
It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of $\text{\hspace{0.17em}}\mathrm{30\xb0},\mathrm{60\xb0},$ and $\text{\hspace{0.17em}}\mathrm{45\xb0}.\text{\hspace{0.17em}}$ Remember, however, that when dealing with right triangles, we are limited to angles between $\text{\hspace{0.17em}}\mathrm{0\xb0}\text{and90\xb0}\text{.}$
Suppose we have a $\text{\hspace{0.17em}}\mathrm{30\xb0},\mathrm{60\xb0},\mathrm{90\xb0}\text{\hspace{0.17em}}$ triangle, which can also be described as a $\text{\hspace{0.17em}}\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2}\text{\hspace{0.17em}}$ triangle. The sides have lengths in the relation $\text{\hspace{0.17em}}s,\text{s}\sqrt{3},2s.\text{\hspace{0.17em}}$ The sides of a $\text{\hspace{0.17em}}\mathrm{45\xb0},\mathrm{45\xb0},\mathrm{90\xb0}\text{\hspace{0.17em}}$ triangle, which can also be described as a $\text{\hspace{0.17em}}\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{2}\text{\hspace{0.17em}}$ triangle, have lengths in the relation $\text{\hspace{0.17em}}s,s,\sqrt{2}s.\text{\hspace{0.17em}}$ These relations are shown in [link] .
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.
Given trigonometric functions of a special angle, evaluate using side lengths.
Find the exact value of the trigonometric functions of $\text{\hspace{0.17em}}\frac{\pi}{3},$ using side lengths.
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