How to find the period of the tan graph?
Understand the Problem
The question is asking how to determine the period of the tangent graph in trigonometry. The period of a trigonometric function is the length of one complete cycle of the wave, and for the tangent function, it involves understanding its properties and behavior.
Answer
The period of the tangent function is $\pi$, which changes to $\frac{\pi}{|k|}$ for $y = \tan(kx)$.
Answer for screen readers
The period of the tangent function is $\pi$, and for $y = \tan(kx)$ it is $\frac{\pi}{|k|}$.
Steps to Solve
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Identify the tangent function The tangent function can be expressed in terms of sine and cosine: $$ y = \tan(x) = \frac{\sin(x)}{\cos(x)} $$
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Understand the basic period of the tangent function The basic period of the tangent function is currently defined as the interval over which the function completes one full cycle. The tangent function has a period of: $$ \pi $$
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Analyze the effect of any transformations If the tangent function is in the form $y = \tan(kx)$, the period changes to: $$ \text{Period} = \frac{\pi}{|k|} $$ where $k$ is a coefficient affecting the input $x$.
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Example of finding the period for $y = \tan(2x)$ For the function $y = \tan(2x)$, we substitute $k = 2$: $$ \text{Period} = \frac{\pi}{|2|} = \frac{\pi}{2} $$
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Conclude the results The period of the tangent function is essential for understanding its behavior and plotting it on the coordinate plane.
The period of the tangent function is $\pi$, and for $y = \tan(kx)$ it is $\frac{\pi}{|k|}$.
More Information
The period of the tangent function indicates how often the function repeats its values. Understanding this helps in graphing the function accurately, especially in terms of identifying asymptotes and the repeating pattern of the curve.
Tips
- Confusing the period of tangent with that of sine or cosine, which is $2\pi$. To avoid this, remember that tangent has a unique period of $\pi$.
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