Reciprocal Trigonometric Functions

We first explore the reciprocal trigonometric functions by studying the relationships between side lengths in a right triangle.

Recall that the trigonometric functions relate the angles in a right triangle to the ratios of the sides. Given the following triangle:

with \( 0^\circ < \theta < \frac{\pi}{2}, \) we have the basic trigonometric functions

\[\begin{array} &\sin \theta = \frac{b}{c}, & \cos \theta = \frac{a}{c}, & \tan \theta = \frac{b}{a} \end{array}\]

and the reciprocal trigonometric functions

\[\begin{align} \csc \theta &= \frac{1}{\sin \theta} = \frac{c}{b} \\\\ \sec \theta &= \frac{1}{\cos \theta} = \frac{c}{a} \\\\ \cot \theta &= \frac{1}{\tan \theta} = \frac{a}{b}. \end{align}\]

We can also connect the reciprocal trigonometric functions to the unit circle, similar to the way we connected the basic trigonometric functions to the unit circle:

Unit Circle

We have

\[\begin{align} \csc \theta &= \frac{1}{\sin \theta} = \frac{\text{radius}}{y} = \frac{1}{y} \\\\ \sec \theta &= \frac{1}{\cos \theta} = \frac{\text{radius}}{x} = \frac{1}{x} \\\\ \cot \theta &= \frac{1}{\tan \theta} = \frac{x}{y}. \end{align}\]

There are certain values for the reciprocal trigonometric functions which are useful to remember:

\[ \begin{array} {| c | c | c | c | c | c |} \hline \theta & 0^\circ & \frac{\pi}{6} = 30^\circ & \frac{\pi}{4} = 45^\circ & \frac{\pi}{3} = 60^\circ & \frac{\pi}{2} = 90^\circ\\ \hline \csc \theta & \infty & \frac {2}{\sqrt{1}} & \frac{2}{\sqrt{2}} & \frac{2}{\sqrt{3}} & \frac{2} {\sqrt{4}} \\ \hline \sec \theta & \frac {2} {\sqrt{4}}& \frac {2} {\sqrt{3}} & \frac{2}{\sqrt{2}} & \frac{2}{\sqrt{1}} & \infty\\ \hline \cot \theta & \infty & \sqrt{3} & 1 & \frac{1}{\sqrt{3}} & 0 \\ \hline \end{array}\]

By remembering the specific values of basic trigonometric functions and the relationships between the basic and reciprocal functions above, these specific values are readily recalled.

We have seen that the basic trigonometric functions have the following behavior in the four quadrants of the plane:

Since all of the basic trigonometric functions are positive in the first quadrant, the reciprocal trigonometric functions are also positive in the first quadrant. Similarly, we have the behavior for the reciprocal trigonometric functions in the remaining quadrants of the plane:

What are the values of \(\theta\) in the range \( 0 \leq \theta < 2 \pi \) such that \(\cot \theta = 1?\)

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From the unit circle visualization above, we see that \(\theta = \frac{\pi}{4} \)and \(\theta = \frac{5\pi}{4} \) satisfy

\[\begin{align} \sin \left( \frac{\pi}{4} \right) &= \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\\ \sin \left( \frac{5\pi}{4} \right) &= \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2}. \end{align}\]

Therefore, these values satisfy \(\cot \theta = \frac{\cos \theta}{\sin \theta} = 1\). We also observe that the line \(y=x\) intersects the unit circles for only these two values of \(\theta\), so the values of \(\theta\) satisfying the required conditions are \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}. \) \(_\square\)

For which values of \(\theta\) in the range \(0 \leq \theta < 2\pi\) are the functions \(\csc \theta, \sec \theta, \) and \(\cot \theta\) not defined?

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From the above definition of \(\csc \theta, \sec \theta, \) and \(\cot \theta,\) we see that \(\csc \theta = \frac{1}{\sin \theta}\) is not defined when \(\sin \theta = 0\), which occurs for \(\theta = 0, \pi\).

Similarly, \(\sec \theta = \frac{1}{\cos \theta}\) is undefined for \(\cos \theta = 0\), which occurs when \(\theta = \frac{\pi}{2}, \frac{3\pi}{2};\) \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) is undefined for \(\sin \theta = 0\), which occurs when \(\theta = 0, \pi.\) \(_\square\)