How to find the period of tan?

Understand the Problem

The question is asking how to determine the period of the tangent function. The period of a function is the interval after which the function repeats its values. To find the period of the tangent function, we look for the interval of x for which tan(x) repeats its values.

Answer

The period of the tangent function is $\pi$.
Answer for screen readers

The period of the tangent function is $\pi$.

Steps to Solve

  1. Identify the function

The tangent function is given by $y = \tan(x)$.

  1. Understand the properties of the tangent function

The tangent function is defined as the ratio of the sine and cosine functions: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$

  1. Determine when the function is undefined

The tangent function is undefined when the cosine is zero. This occurs at the angles where: $$ \cos(x) = 0 $$ These points are at $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer.

  1. Calculate the period

The period of the tangent function can be found by looking at the difference between consecutive points where the function is defined. The points are: $$ x_1 = \frac{\pi}{2} + n\pi \ x_2 = \frac{\pi}{2} + (n+1)\pi $$ The difference is: $$ x_2 - x_1 = \left( \frac{\pi}{2} + (n+1)\pi \right) - \left( \frac{\pi}{2} + n\pi \right) = \pi $$

  1. Conclusion

Thus, the period of the tangent function is $\pi$.

The period of the tangent function is $\pi$.

More Information

The tangent function repeats its values every $\pi$ radians. This is different from the sine and cosine functions, which have a period of $2\pi$.

Tips

  • Confusing the period of tangent with that of sine or cosine functions.
  • Forgetting that tangent is undefined at certain angles, which can affect its graph and period.

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