What is the period for the tangent function?
Understand the Problem
The question is asking for the period of the tangent function in mathematics. The period refers to the interval over which the function repeats its values. For the tangent function, we understand that it has distinct properties related to periodicity.
Answer
The period of the tangent function is $\pi$.
Answer for screen readers
The period of the tangent function is $\pi$.
Steps to Solve
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Identify the tangent function The tangent function is represented as $tan(x)$.
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Understand periodicity of the function The tangent function is periodic, which means it repeats its values over a specific interval.
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Determine the period of the tangent function The period of $tan(x)$ can be derived from its relationship to the sine and cosine functions. The tangent can be expressed as: $$ tan(x) = \frac{sin(x)}{cos(x)} $$ Since cosine has zeros (where the function is undefined) at $x = \frac{\pi}{2} + n\pi$ (for any integer $n$), we can derive the period.
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Calculate the period The tangent function completes one full cycle from $x = 0$ to $x = \pi$, thus the period is: $$ \text{Period} = \pi $$
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Conclusion Hence, the period of the tangent function is established as $\pi$.
The period of the tangent function is $\pi$.
More Information
The tangent function is important in trigonometry and is often used in various applications, including physics and engineering. Its unique periodic properties allow it to model various waveforms and oscillatory behaviors.
Tips
- Confusing the period of the tangent function with the period of the sine or cosine function, which is $2\pi$. To avoid this mistake, remember that the tangent function has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.
