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
- Identify the function
The tangent function is given by $y = \tan(x)$.
- Understand the properties of the tangent function
The tangent function is defined as the ratio of the sine and cosine functions: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
- 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.
- 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: $$ x_2 - x_1 = \left( \frac{\pi}{2} + (n+1)\pi \right) - \left( \frac{\pi}{2} + n\pi \right) = \pi $$
- 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.