Learning objective
Recall that the cosecant function for the central angle, \theta, on the unit circle, is written as \csc \theta and it is defined by \csc \theta = \dfrac{1}{\sin \theta}. Similarly, the secant function is defined by \sec \theta = \dfrac{1}{\cos \theta}. Lastly, the cotangent function is defined by \cot \theta= \dfrac{1}{\tan \theta}.
Explore the applet by checking the boxes.
The reciprocal trigonometric functions are related to the exact values on the unit circle like the basic trigonometric functions are related to the unit circle, so the graphs of the reciprocal functions are also related to the graphs of the basic trigonometric functions.
Some of the key features of the cosecant function are as follows:
Some of the key features of the secant function are as follows:
Some of the key features of the cotangent function are as follows:
Sketch the graph of f \left( \theta \right) = \sec \theta on the interval \left[ -2 \pi, 2 \pi \right].
Consider the function f \left(x\right) =\csc \left(3x\right).
State the equations of the asymptotes on the interval \left[0, 2\pi\right].
Sketch the graph of f \left(x\right) = \csc \left(3x\right).
Explain whether the following key features are affected by the transformation:
Consider the function f \left( \theta \right) = \cot \theta and the following table of values:
\theta | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
---|---|---|---|---|---|---|---|---|---|
\cos \theta | 1 | \dfrac{\sqrt{2}}{2} | 0 | -\dfrac{\sqrt{2}}{2} | -1 | -\dfrac{\sqrt{2}}{2} | 0 | \dfrac{\sqrt{2}}{2} | 1 |
\sin \theta | 0 | \dfrac{\sqrt{2}}{2} | 1 | \dfrac{\sqrt{2}}{2} | 0 | -\dfrac{\sqrt{2}}{2} | -1 | -\dfrac{\sqrt{2}}{2} | 0 |
\cot \theta |
State the values of \theta on the interval \left[0, 2\pi\right] for which f \left( \theta \right) is undefined.
Sketch the graph of f \left( \theta \right) = \cot \theta on the interval \left[ -2 \pi, 2 \pi \right].
Compare the domain and range of f \left( \theta \right) = \cot \theta to g \left( \theta \right) = \tan \theta.
It will be useful to recall the relationship between the graphs of the three basic trigonometric functions and their reciprocal graphs: