Find The Equation Of A Sine Or Cosine Graph
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These lessons are compiled to help Algebra 2 students find the equations of sine and cosine graphs.
The equations of a sine graph and a cosine graph can be expressed in general forms that accounts for various transformations like stretches, shifts, and reflections.
The following diagram shows how to find the equation of a sine graph. Scroll down the page for more examples and solutions.
General Equations:
Sine: y = A sin(B(x - D)) + C
Cosine: y = A cos(B(x - D)) + C
These equations look very similar, and that’s because sine and cosine are essentially the same graph shifted horizontally.
Let’s break down what each part of these equations represents:
A (Amplitude):
The amplitude is the vertical distance from the midline of the graph to its peak (maximum) or trough (minimum).
|A| (the absolute value of A) represents the amplitude.
If A is negative, the graph is reflected across the x-axis.
B (Period):
The period is the horizontal distance it takes for the graph to complete one full cycle (from peak to peak or trough to trough).
The period is calculated as: Period = 2π / |B|
C (Vertical Shift or Midline):
The vertical shift is the vertical displacement of the graph. It determines the midline (horizontal center line) of the graph.
If C is positive, the graph shifts up.
If C is negative, the graph shifts down.
D (Phase Shift or Horizontal Shift):
The phase shift is the horizontal displacement of the graph.
If D is positive, the graph shifts to the right.
If D is negative, the graph shifts to the left.
Key Differences Between Sine and Cosine:
The main difference between the sine and cosine graphs is their starting point:
y = sin(x): Starts at the origin (0, 0).
y = cos(x): Starts at its maximum value on the y-axis (0, 1).
This means that the cosine graph is essentially a sine graph shifted π/2 units to the left.
cos(x) = sin(x + π/2)
Find An Equation For The Sine Or Cosine Wave
When finding the equation for a trig function, try to identify if it is a sine or cosine graph.
To find the equation of sine waves given the graph:
- Find the amplitude which is half the distance between the maximum and minimum.
- Find the period of the function which is the horizontal distance for the function to repeat. If the period is more than 2π then B is a fraction; use the formula period = 2π/B to find the exact value.
- Find any phase shift, h.
How To Determine The Equation Of A Sine And Cosine Graph?
The general equation of a sine graph is y = A sin(B(x - D)) + C
The general equation of a cosine graph is y = A cos(B(x - D)) + C
Examples:
Given a transformed graph of sine or cosine, determine a possible equation.
Trigonometry: Finding An Equation From A Graph
Writing Equation Of Sin And Cos Graph
Examples:
Find the equation of the given graph in terms of sine and cosine.
Determining The Equation Of A Trigonometric Function
Determining the amplitude and period of sine and cosine functions.
How To Come Up With The Equation Of A Sin/Cos Function When Given The Graph?
How To Get Equation Of A Sine Curve?
How To Identify The Graph Of A Stretched Cosine Curve?
This video demonstrates how to stretch and shrink the sine and cosine curves.
Finding The Equation Of Sine Or Cosine Given Its Graph
Sine Graph Calculator
We have included a tool that will plot the sine graph f(x) = A sin(B(x-h))+ k, given the values A, B, h and k. Use it to check your answers.
Enter in the values for f(x) = Asin(B(x-h))+k into the sine graph calculator to check your answer.
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
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