Learning objectives
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Similarly to the graphs of sine and cosine, the graph of the tangent function is related to the exact values of tangent on the unit circle.
The tangent function, f \left( \theta \right)= \tan \theta, is a periodic function with a domain represented by the measure \theta of an angle in standard position, and a range, \tan \theta, the quotient of the x- and y-coordinates, \dfrac{y}{x}, of the right triangle positioned on the unit circle.
Because \tan{\theta}= \dfrac{\sin{\theta}}{\cos{\theta}}, the tangent function is undefined anywhere \cos{\theta}=0. This results in the tangent function being a discontinuous function with a period of \pi.
Consider the function f \left( \theta \right) = \tan \theta.
Complete the table with values in exact form:
\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 |
---|---|---|---|---|---|---|---|---|---|
\tan \theta |
Sketch a graph for f \left( \theta \right) = \tan \theta on the domain [-2\pi, 2\pi].
Consider the graph of f \left( \theta \right) = \tan \theta:
Describe the behavior of the graph as \theta \to \dfrac{\pi}{2} from the left. Explain how this relates to the unit circle.
Describe the behavior of the graph as \theta \to \pi from the left. Explain how this relates to the unit circle.
Identify the positive and negative intervals on the graph of f \left( \theta \right) = \tan \theta over the domain [0, 2 \pi]. Explain how this relates to the unit circle.
Because \tan{\theta}= \dfrac{\sin{\theta}}{\cos{\theta}}, the tangent function is undefined anywhere \cos{\theta}=0. This results in the tangent function being a discontinuous function with a period of \pi.
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The graph of the tangent function can be transformed by adjusting its frequency and midline. Note that the tangent function does not have an amplitude because it does not have a maximum or minimum value. However, we can change the steepness of the function.
Consider the transformed graph of f \left( \theta \right) = \tan \theta:
The given graph is the result of what transformations on f \left( \theta \right) = \tan \theta?
State the equation of the graphed function.
State the domain and range.
Consider the function f \left( \theta \right) = -\tan \left( \theta + \dfrac{\pi}{2} \right).
Graph the function over the domain \left[-2 \pi, 2 \pi\right].
Identify the key features of f \left( \theta \right) = -\tan \left( \theta + \dfrac{\pi}{2} \right).
Compare the graph of f \left( \theta \right) = \tan \theta to each function.
g \left( \theta \right) = \tan \dfrac{1}{2} \theta
h \left( \theta \right) = 6 \tan \theta
Refer to the general form of the tangent function to write or determine transformations to f \left( \theta \right) = \tan \theta: