What is the Period of Sine Function?

S6-SA2-0323

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The period of a sine function is the length of one complete cycle of the wave. It's the smallest positive value after which the graph of the sine function starts repeating itself exactly.

Simple Example

Quick Example

Imagine a cricket ball swinging back and forth. If it takes 2 seconds to complete one full swing (from one side to the other and back to the start), then 2 seconds is the 'period' of its swing. The sine function also repeats its pattern after a certain 'time' or 'interval'.

Worked Example

Step-by-Step

Let's find the period of the function y = sin(2x).

1. The standard sine function is y = sin(Bx), where B is a number.
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2. For the standard sine function y = sin(x), the period is 2pi.
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3. For a function y = sin(Bx), the period is calculated using the formula: Period = 2pi / |B|.
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4. In our function y = sin(2x), the value of B is 2.
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5. Substitute B = 2 into the formula: Period = 2pi / |2|.
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6. Calculate the value: Period = 2pi / 2 = pi.
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Answer: The period of y = sin(2x) is pi.

Why It Matters

Understanding periods helps engineers design structures that can withstand repeating forces, like bridges dealing with traffic vibrations. In Physics, it's crucial for understanding sound waves and light waves. Even AI/ML models use periodic functions to analyze repeating patterns in data, helping predict future trends.

Common Mistakes

MISTAKE: Thinking the period is always 2pi, regardless of the 'B' value inside sin(Bx). | CORRECTION: The period is 2pi ONLY for sin(x). For sin(Bx), you must use the formula Period = 2pi / |B|.

MISTAKE: Forgetting the absolute value in the formula, leading to negative periods. | CORRECTION: The period is a length or an interval, so it must always be a positive value. Use Period = 2pi / |B|.

MISTAKE: Confusing the amplitude with the period. | CORRECTION: Amplitude is the height of the wave from its center line, while the period is the horizontal length of one complete wave cycle.

Practice Questions

Try It Yourself

QUESTION: What is the period of the function y = sin(4x)? | ANSWER: pi/2

QUESTION: If the period of a sine function is 4pi, what could be the function? | ANSWER: y = sin(x/2)

QUESTION: A sound wave can be modeled by y = 5sin(0.5x). What is its period? | ANSWER: 4pi

MCQ

Quick Quiz

What is the period of the sine function y = sin(x/3)?

2pi

pi/3

6pi

3pi

The Correct Answer Is:

The formula for the period is 2pi / |B|. Here, B = 1/3. So, Period = 2pi / (1/3) = 2pi * 3 = 6pi. Therefore, option C is correct.

Real World Connection

In the Real World

Think about the alternating current (AC) electricity that powers your home appliances like fans and lights. The voltage in an AC circuit follows a sine wave pattern. The 'period' tells us how often the current completes one cycle, which is important for designing electrical grids and devices in India.

Key Vocabulary

Key Terms

PERIOD: The length of one complete cycle of a repeating function | SINE FUNCTION: A type of wave-like mathematical function | CYCLE: One complete repetition of a pattern | AMPLITUDE: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.

What's Next

What to Learn Next

Now that you understand the period, explore the 'Amplitude of Sine Function'. Knowing both period and amplitude will help you fully describe and graph sine waves, which is super useful in higher classes for Physics and Engineering.