How to Find Period of a Sine Function
Understanding the period of a sine function is fundamental in trigonometry and essential for analyzing periodic phenomena in mathematics, physics, and engineering. For the basic sine function ( y = \sin(x) ), the period is ( 2\pi ). Still, when the function includes a coefficient ( B ) inside the sine term, such as ( y = \sin(Bx) ), the period changes to ( \frac{2\pi}{|B|} ). The period of a sine function refers to the horizontal length required for the function to complete one full cycle and return to its starting value. This guide will walk you through the steps to determine the period of any sine function, supported by examples and explanations.
Steps to Find the Period of a Sine Function
Step 1: Identify the General Form of the Function
The general form of a sine function is:
[ y = A \sin(Bx + C) + D ]
Here:
- ( A ) = Amplitude (vertical stretch/compression)
- ( B ) = Frequency coefficient (affects the period)
- ( C ) = Phase shift (horizontal shift)
- ( D ) = Vertical shift
Focus on identifying the coefficient ( B ) in the function. This value directly influences the period.
Step 2: Apply the Period Formula
The formula to calculate the period ( P ) of a sine function is:
[ P = \frac{2\pi}{|B|} ]
- Important: Always use the absolute value of ( B ) to ensure a positive period.
Step 3: Simplify the Expression
Substitute the value of ( B ) into the formula and simplify.
Example 1: Basic Function
Function: ( y = \sin(3x) )
- Here, ( B = 3 ).
- Period: ( P = \frac{2\pi}{3} ).
Example 2: Negative Coefficient
Function: ( y = \sin(-4x) )
- ( B = -4 ), but the absolute value is ( 4 ).
- Period: ( P = \frac{2\pi}{4} = \frac{\pi}{2} ).
Example 3: Fractional Coefficient
Function: ( y = \sin\left(\frac{1}{2}x\right) )
- ( B = \frac{1}{2} ).
- Period: ( P = \frac{2\pi}{\frac{1}{2}} = 4\pi ).
Scientific Explanation: Why Does the Formula Work?
The sine function ( \sin(x) ) completes one full cycle over an interval of ( 2\pi ). When the input ( x ) is multiplied by ( B ), it compresses or stretches the graph horizontally:
- If ( |B| > 1 ), the graph is compressed, shortening the period.
- If ( |B| < 1 ), the graph is stretched, lengthening the period.
Mathematically, the argument ( Bx ) changes at a different rate than ( x ). Here's one way to look at it: in ( y = \sin(2x) ), the input ( 2x
