How to Find the Period of Trigonometric and Other Functions
The concept of period of a function plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether analyzing waves, solving trigonometry problems, or tackling JEE and CBSE questions, understanding how a function repeats over intervals helps master many types of problems with confidence.
What Is Period of a Function?
A period of a function is the smallest positive interval after which a function repeats its values. If f(x) = f(x + P) for all x and some positive number P, then P is called the period. You’ll find this concept applied in areas such as trigonometric functions, physics (like oscillations and waves), and computer science (signals and patterns).
Key Formula for Period of a Function
Here’s the standard formula for periodic trigonometric functions:
For \( f(x) = \sin(ax) \) or \( f(x) = \cos(ax) \):
Period = \( \frac{2\pi}{|a|} \)
For \( f(x) = \tan(ax) \):
Period = \( \frac{\pi}{|a|} \)
| Function | Standard Period |
|---|---|
| sin(x) | 2π |
| cos(x) | 2π |
| tan(x) | π |
Cross-Disciplinary Usage
The period of a function is not only useful in Maths but also plays an important role in Physics (waves, oscillations), Computer Science (signal processing), and daily logical reasoning. Students preparing for JEE or NEET will often encounter questions where identifying or calculating the period is essential for quick solutions.
Step-by-Step Illustration
Let’s find the period of the function \( f(x) = \sin(3x) \):
1. Note the standard sine function has a period of 2π.2. Our function is \( \sin(ax) \) with a = 3.
3. Use the formula: Period = \( \frac{2\pi}{|a|} = \frac{2\pi}{3} \).
4. Final Answer: The period is \( \frac{2\pi}{3} \).
How to Find the Period: Graphical and Algebraic
You can also find the period of a function by observing its graph. Notice the distance along the x-axis after which the shape starts repeating. In equation-based questions, always look for the coefficient beside x and apply the formula. For complex or composite functions, use the lowest common multiple (LCM) of their individual periods.
Speed Trick or Vedic Shortcut
Here’s a handy trick to instantly recall periods for standard trigonometric functions:
- For sin(ax) and cos(ax): Divide 2π by the coefficient of x.
- For tan(ax): Divide π by the coefficient of x.
Quick Example: What’s the period of \( f(x) = \cos(5x) \)?
Just apply: Period = 2π/5.
Students in Vedantu’s live online sessions practice such shortcuts to save time in competitive exams.
Try These Yourself
- Find the period of \( f(x) = \tan(2x) \).
- Check: Is \( f(x) = x^2 \) a periodic function?
- What is the period of \( g(x) = \sin(4x + 1) \)?
- Can the sum of two periodic functions with different periods be periodic?
Frequent Errors and Misunderstandings
- Confusing period with amplitude (height vs. repeating interval).
- Forgetting to use the absolute value in the denominator of the formula.
- Assuming all functions are periodic (not true for, say, linear or quadratic functions).
Period, Amplitude, and Frequency—What’s the Difference?
| Concept | Meaning |
|---|---|
| Period | Time or interval after which a function repeats. |
| Amplitude | Maximum "height" or value from the mean. |
| Frequency | Number of cycles completed in one second. Frequency = 1/Period. |
Relation to Other Concepts
The idea of period of a function connects closely with topics such as amplitude and period, composite and inverse functions, and trigonometric functions. Mastering this helps you with graph transformations, solving trigonometric equations, and analyzing data patterns.
Classroom Tip
A good way to remember the period of a function for sine/cosine is to look for the "b" in \( f(x) = \sin(bx) \): divide 2π by b. Draw graphs with colored pencils to visually spot one complete cycle. Vedantu’s teachers use this strategy for better clarity during interactive sessions.
We explored period of a function—from the definition, formula, graphing tips, example problems, and common mistakes to helpful exam strategies. Continue practicing with Vedantu and explore other related topics to build a strong foundation for Maths success.
Related Vedantu Pages for Deeper Understanding
FAQs on Period of a Function in Maths: Definition, Formula & Examples
1. What is the period of a function in Maths?
In mathematics, the period of a periodic function is the smallest positive value P such that f(x + P) = f(x) for all x in the domain of the function. In simpler terms, it's the interval after which the function's values repeat themselves. For example, the sine function, sin(x), has a period of 2π because its values repeat every 2π units.
2. How do you calculate the period from a trigonometric equation?
The method for calculating the period depends on the specific trigonometric function. For functions of the form f(x) = Asin(Bx + C) or f(x) = Acos(Bx + C), the period is calculated as Period = 2π/|B|. For tangent functions, the period is π/|B|. The absolute value ensures a positive period. Remember that B is the coefficient of x within the trigonometric function.
3. What is the period of sin(x) and cos(x)?
Both the sine function, sin(x), and the cosine function, cos(x), have a period of 2π. This means their values repeat every 2π units along the x-axis.
4. What happens to the period if the function is stretched or compressed?
Stretching or compressing a trigonometric function affects its period. If the function is of the form f(x) = sin(Bx) or cos(Bx), then:
• A larger value of |B| (|B| > 1) compresses the function, resulting in a shorter period.
• A smaller value of |B| (0 < |B| < 1) stretches the function, resulting in a longer period. The period is inversely proportional to |B|.
5. How can I find the period from a graph?
To find the period from a graph of a periodic function, identify any two consecutive corresponding points where the graph repeats itself. These could be consecutive peaks, troughs, or any other matching points. The horizontal distance between these points represents the period of the function.
6. What is the period of tan(x)?
The tangent function, tan(x), has a period of π. Unlike sine and cosine, its graph repeats every π units.
7. How do you find the period of a composite function f(ax + b)?
For a composite function of the form f(ax + b), where f(x) has a period P, the period of the composite function is given by P/|a|. The constant b does not affect the period.
8. What is the difference between the period and the amplitude?
The period refers to the horizontal distance over which a function repeats its values. The amplitude, on the other hand, refers to the vertical distance from the center line of the function to its maximum or minimum value. They are independent characteristics of a periodic function.
9. What is the relationship between period and frequency?
Period and frequency are inversely related. Frequency (f) is the number of cycles per unit time, while the period (T) is the time taken for one complete cycle. Their relationship is expressed as: f = 1/T or T = 1/f.
10. Can a function have multiple periods?
While a function has a fundamental period (the smallest period), it can also have multiple periods. Any integer multiple of the fundamental period is also a period of the function. However, it is the fundamental period that is typically of interest.
11. How is the concept of period used in physics and engineering?
The concept of period is crucial in physics and engineering for describing oscillatory or periodic phenomena, such as simple harmonic motion, wave propagation, and alternating currents. Understanding the period helps in analyzing the frequency and characteristics of these systems.
12. What are some real-life examples of periodic functions?
Real-life examples of periodic functions include:
• The oscillation of a pendulum
• Sound waves
• Alternating current (AC) electricity
• The rhythmic beating of a heart. These all exhibit repeating patterns over time, making them describable using periodic functions.
