Reciprocal Functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.

Take another look at these definitions. These functions are the reciprocals of the first three functions.

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 (Figure). The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

Figure 5. The side adjacent to one angle is opposite the other angle.

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.

Evaluating Trigonometric Functions of Angles Not in Standard Position

Using the triangle shown in (Figure), evaluate[latex]\,\mathrm{sin}\,\alpha ,\mathrm{cos}\,\alpha ,\mathrm{tan}\,\alpha ,\mathrm{sec}\,\alpha ,\mathrm{csc}\,\alpha ,\text{and}\,\mathrm{cot}\,\alpha .[/latex]

Figure 6.

Show Solution

[latex]\begin{array}{ccc}\hfill \text{sin }\alpha & =\frac{\text{opposite }\alpha }{\text{hypotenuse}}\hfill & =\frac{4}{5}\hfill \\ \hfill \text{cos }\alpha & =\frac{\text{adjacent to }\alpha }{\text{hypotenuse}}\hfill & =\frac{3}{5}\hfill \\ \hfill \text{tan }\alpha & =\frac{\text{opposite }\alpha }{\text{adjacent to }\alpha }\hfill & =\frac{4}{3}\hfill \\ \hfill \text{sec }\alpha & =\frac{\text{hypotenuse}}{\text{adjacent to }\alpha }\hfill & =\frac{5}{3}\hfill \\ \hfill \text{csc }\alpha & =\frac{\text{hypotenuse}}{\text{opposite }\alpha }\hfill & =\frac{5}{4}\hfill \\ \hfill \text{cot }\alpha & =\frac{\text{adjacent to }\alpha }{\text{opposite }\alpha }\hfill & =\frac{3}{4}\hfill \end{array}[/latex]

Analysis

Another approach would have been to find sine, cosine, and tangent first. Then find their reciprocals to determine the other functions.

[latex]\text{sec }\alpha =\frac{1}{\text{cos }\alpha }=\frac{1}{\frac{3}{5}}=\frac{5}{3}[/latex]

[latex]\text{csc }\alpha =\frac{1}{\text{csc }\alpha }=\frac{1}{\frac{4}{5}}=\frac{5}{4}[/latex]

[latex]\text{cot }\alpha =\frac{1}{\text{tan }\alpha }=\frac{1}{\frac{4}{3}}=\frac{3}{4}[/latex]

Try It

Using the triangle shown in (Figure),evaluate[latex]\,\text{sin}\,t,\text{cos}\,t,\text{tan}\,t,\text{sec}\,t,\text{csc}\,t,\text{and}\,\text{cot}\,t.[/latex]

Figure 7.

Show Solution

[latex]\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}[/latex]

Finding Trigonometric Functions of Special Angles Using Side Lengths

It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of[latex]\,30°,60°,[/latex]and[latex]\,45°.\,[/latex]Remember, however, that when dealing with right triangles, we are limited to angles between[latex]\,0°\text{ and 90°}\text{.}[/latex]

Suppose we have a[latex]\,30°,60°,90°\,[/latex]triangle, which can also be described as a[latex]\,\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2}\,[/latex]triangle. The sides have lengths in the relation[latex]\,s,\sqrt{3}s,2s.\,[/latex]The sides of a[latex]\,45°,45°,90°\,[/latex]triangle, which can also be described as a[latex]\,\frac{\pi }{4},\frac{\pi }{4},\frac{\pi }{2}\,[/latex]triangle, have lengths in the relation[latex]\,s,s,\sqrt{2}s.\,[/latex]These relations are shown in (Figure).

Figure 8. Side lengths of special triangles

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

How To

Given trigonometric functions of a special angle, evaluate using side lengths.

1. Use the side lengths shown in (Figure) for the special angle you wish to evaluate.
2. Use the ratio of side lengths appropriate to the function you wish to evaluate.

Evaluating Trigonometric Functions of Special Angles Using Side Lengths

Find the exact value of the trigonometric functions of[latex]\,\frac{\pi }{3},[/latex]using side lengths.

Show Solution

[latex]\begin{array}{ccccc}\hfill \mathrm{sin}\left(\frac{\pi }{3}\right)& =\frac{\text{opp}}{\text{hyp}}\hfill & =\frac{\sqrt{3s}}{2s}\hfill & =\frac{\sqrt{3}}{2}\hfill & \\ \hfill \mathrm{cos}\left(\frac{\pi }{3}\right)& =\frac{\text{adj}}{\text{hyp}}\hfill & =\frac{s}{2s}\hfill & =\frac{1}{2}\hfill & \\ \hfill \mathrm{tan}\left(\frac{\pi }{3}\right)& =\frac{\text{opp}}{\text{adj}}\hfill & =\frac{\sqrt{3}s}{s}\hfill & =\sqrt{3}\hfill & \\ \hfill \mathrm{sec}\left(\frac{\pi }{3}\right)& =\frac{\text{hyp}}{\text{adj}}\hfill & =\frac{2s}{s}\hfill & =2\hfill & \\ \hfill \mathrm{csc}\left(\frac{\pi }{3}\right)& =\frac{\text{hyp}}{\text{opp}}\hfill & =\frac{2s}{\sqrt{3}s}\hfill & =\frac{2}{\sqrt{3}}\hfill & =\frac{2\sqrt{3}}{3}\hfill \\ \hfill \mathrm{cot}\left(\frac{\pi }{3}\right)& =\frac{\text{adj}}{\text{opp}}\hfill & =\frac{s}{\sqrt{3}s}\hfill & =\frac{1}{\sqrt{3}}\hfill & =\frac{\sqrt{3}}{3}\hfill \end{array}[/latex]

Try It

Find the exact value of the trigonometric functions of[latex]\,\frac{\pi }{4},[/latex]using side lengths.

Show Solution

[latex]\begin{array}{ccc}\hfill \text{sin}\left(\frac{\pi }{4}\right)& =& \frac{\sqrt{2}}{2},\text{cos}\left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2},\mathrm{tan}\left(\frac{\pi }{4}\right)=1,\hfill \\ \hfill \text{sec}\left(\frac{\pi }{4}\right)& =& \sqrt{2},\text{csc}\left(\frac{\pi }{4}\right)=\sqrt{2},\text{cot}\left(\frac{\pi }{4}\right)=1\hfill \end{array}[/latex]

Using Equal Cofunction of Complements

If we look more closely at the relationship between the sine and cosine of the special angles, we notice a pattern. In a right triangle with angles of[latex]\,\frac{\pi }{6}\,[/latex]and[latex]\,\frac{\pi }{3},[/latex]we see that the sine of[latex]\,\frac{\pi }{3},[/latex]namely[latex]\,\frac{\sqrt{3}}{2},[/latex]is also the cosine of[latex]\,\frac{\pi }{6},[/latex]while the sine of[latex]\,\frac{\pi }{6},[/latex]namely[latex]\,\frac{1}{2},[/latex]is also the cosine of[latex]\,\frac{\pi }{3}.[/latex]

See (Figure).

Figure 9. The sine of[latex]\,\frac{\pi }{3}\,[/latex]equals the cosine of[latex]\,\frac{\pi }{6}\,[/latex]and vice versa.

This result should not be surprising because, as we see from (Figure), the side opposite the angle of[latex]\,\frac{\pi }{3}\,[/latex]is also the side adjacent to[latex]\,\frac{\pi }{6},[/latex]so[latex]\,\mathrm{sin}\left(\frac{\pi }{3}\right)\,[/latex]and[latex]\,\mathrm{cos}\left(\frac{\pi }{6}\right)\,[/latex]are exactly the same ratio of the same two sides,[latex]\,\sqrt{3}s\,[/latex]and[latex]\,2s.\,[/latex]Similarly,[latex]\,\mathrm{cos}\left(\frac{\pi }{3}\right)\,[/latex]and[latex]\,\mathrm{sin}\left(\frac{\pi }{6}\right)\,[/latex]are also the same ratio using the same two sides,[latex]\,s\,[/latex]and[latex]\,2s.[/latex]

The interrelationship between the sines and cosines of[latex]\,\frac{\pi }{6}\,[/latex]and[latex]\,\frac{\pi }{3}\,[/latex]also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to[latex]\,\pi ,[/latex]and the right angle is[latex]\,\frac{\pi }{2},[/latex]the remaining two angles must also add up to[latex]\,\frac{\pi }{2}.\,[/latex]That means that a right triangle can be formed with any two angles that add to[latex]\,\frac{\pi }{2}\,[/latex]—in other words, any two complementary angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in (Figure).

Figure 10. Cofunction identity of sine and cosine of complementary angles

Using this identity, we can state without calculating, for instance, that the sine of[latex]\,\frac{\pi }{12}\,[/latex]equals the cosine of[latex]\,\frac{5\pi }{12},[/latex]and that the sine of[latex]\,\frac{5\pi }{12}\,[/latex]equals the cosine of[latex]\,\frac{\pi }{12}.\,[/latex]We can also state that if, for a given angle[latex]\,t,\mathrm{cos}\text{ }t=\frac{5}{13},[/latex]then[latex]\,\mathrm{sin}\left(\frac{\pi }{2}-t\right)=\frac{5}{13}\,[/latex]as well.

Using Trigonometric Functions

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in (Figure).

Figure 11.

Show Solution

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

[latex]\mathrm{tan}\left(30°\right)=\frac{7}{a}[/latex]

We rearrange to solve for[latex]\,a.[/latex]

[latex]\begin{array}{ccc}\hfill a& =& \frac{7}{\mathrm{tan}\left(30°\right)}\hfill \\ & \approx & 12.1\hfill \end{array}[/latex]

We can use the sine to find the hypotenuse.

[latex]\mathrm{sin}\left(30°\right)=\frac{7}{c}[/latex]

Again, we rearrange to solve for[latex]\,c.[/latex]

[latex]\begin{array}{ccc}\hfill c& =& \frac{7}{\mathrm{sin}\left(30°\right)}\hfill \\ & =& 14\hfill \end{array}[/latex]

Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height.

Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. See (Figure).

Figure 12.

Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of[latex]\,57°\,[/latex]between a line of sight to the top of the tree and the ground, as shown in (Figure). Find the height of the tree.

Figure 13.

Show Solution

We know that the angle of elevation is[latex]\,57°\,[/latex]and the adjacent side is 30 ft long. The opposite side is the unknown height.

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of[latex]\,57°,[/latex]letting[latex]\,h\,[/latex]be the unknown height.

[latex]\begin{array}{cccc}\hfill \text{tan }\theta & =& \frac{\text{opposite}}{\text{adjacent}}\hfill & \\ \hfill \text{tan}\left(57°\right)& =& \frac{h}{30}\hfill & \phantom{\rule{1em}{0ex}}\text{Solve for }h.\hfill \\ \hfill h& =& \text{30tan}\left(\text{57°}\right)\hfill & \phantom{\rule{1em}{0ex}}\text{Multiply}\text{.}\hfill \\ \hfill h& \approx & 46.2\hfill & \phantom{\rule{1em}{0ex}}\text{Use a calculator}\text{.}\hfill \end{array}[/latex]

The tree is approximately 46 feet tall.