How to find the period of a tangent function?
Understand the Problem
The question is asking how to determine the period of a tangent function, which is a trigonometric function. The period of a tangent function can be found using the formula for the period of trigonometric functions.
Answer
The period of $y = \tan(kx)$ is $\frac{π}{|k|}$.
Answer for screen readers
The period of the tangent function $y = \tan(kx)$ is given by $\frac{π}{|k|}$, for example, if $k = 2$, the period is $\frac{π}{2}$.
Steps to Solve
- Identify the basic period of the tangent function
The basic period of the tangent function $y = \tan(x)$ is $π$. This means that the function repeats every $π$ radians.
- Understand how the period changes with transformations
If the tangent function is transformed by a coefficient $k$ inside the argument, like so: $y = \tan(kx)$, the period changes. The new period can be found with the formula:
$$ \text{New Period} = \frac{π}{|k|} $$
- Apply the formula with a specific value of ( k )
For example, if we have $y = \tan(2x)$, the value of $k$ is 2. Now, we can plug this into our formula:
$$ \text{New Period} = \frac{π}{|2|} = \frac{π}{2} $$
- Conclusion
Therefore, after calculating, we find the new period for the tangent function with a specific transformation.
The period of the tangent function $y = \tan(kx)$ is given by $\frac{π}{|k|}$, for example, if $k = 2$, the period is $\frac{π}{2}$.
More Information
The period of a function in trigonometry refers to the distance on the $x$-axis it takes for the function to complete one full cycle. For tangent and cotangent, this period is shorter than that of sine and cosine functions.
Tips
- Forgetting to consider the absolute value of ( k ).
- Confusing the tangent function's period with that of sine and cosine, which have periods of $2π$.
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