Find The Equation Of A Sine Or Cosine Graph

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.

 

Trigonometry Worksheets
Practice your skills with the following worksheets:
Printable & Online Trigonometry Worksheets

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.
Effect: Stretches or shrinks the graph vertically. If A is negative, the graph is reflected across the x-axis.
Formula from graph: \(A=\frac{\text{Maximum Value - Minimum Value}}{2}\)

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 \(=\frac{2\pi}{|B|}\)

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 entire graph shifts up.
If C is negative, the entire graph shifts down.
Formula from graph: \(C=\frac{\text{Maximum Value + Minimum Value}}{2}\)

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:

1. Find the amplitude which is half the distance between the maximum and minimum.
2. 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.
3. 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.

• Show Video Lesson

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.

• Show Video Lesson

Determining The Equation Of A Trigonometric Function
Determining the amplitude and period of sine and cosine functions.

• Show Video Lesson

How To Come Up With The Equation Of A Sin/Cos Function When Given The Graph?

• Show Video Lesson

How To Get Equation Of A Sine Curve?

• Show Video Lesson

How To Identify The Graph Of A Stretched Cosine Curve?
This video demonstrates how to stretch and shrink the sine and cosine curves.

• Show Video Lesson

Finding The Equation Of Sine Or Cosine Given Its Graph

• Show Video Lesson

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.

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.