 # Sine and Cosine Graphs

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A series of free online Trigonometry Video Lessons.
Examples, solutions, videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

• how to graph the basic sine and cosine functions
• how to change the amplitude and period of the sine and cosine graphs
• how to transform the sine and cosine graphs using horizontal (phase) shift and vertical shift
• how to graph sine and cosine functions with the four basic transformations: amplitude, period, phase shift and vertical shift.
• how to find the equation of a sine or cosine graph

### Graphs of the Sine and Cosine Functions (Basic Graph)

Sine and cosine are periodic functions, which means that sine and cosine graphs repeat themselves in patterns. You can graph sine and cosine functions by understanding their period and amplitude. Sine and cosine graphs are related to the graph of the tangent function, though the graphs look very different.

How to graph the sine and cosine function on the coordinate plane using the unit circle?
How to determine the domain and range of the sine and cosine function?
How to determine the period of the sine and cosine function?
How to graph the sine and cosine functions by plotting table of values?

### Transforming the Graphs of Sine and Cosine (Change in Amplitude and Period)

The coefficients A and B in y = Asin(Bx) or y = Acos(Bx) each have a different effect on the graph. If A and B are 1, both graphs have an amplitude of 1 and a period of 2pi. For sine and cosine transformations, when A is larger than 1, the amplitude increases and is equal to the value of A; if A is negative, the graph reflects over the x-axis. When B is greater than 1, the period decreases; use the formula 2pi/B to find the period.

Amplitude and Period of Sine and Cosine
This video explains how to determine the amplitude and period of sine and cosine functions. It also shows how to graph the sine and cosine functions with different amplitudes and periods.

### Transformations of Sine and Cosine (Horizontal or Phase shift and Vertical shift)

In the equation y = Asin(B(x-h)) or y = Acos(B(x-h)), A modifies the amplitude and B modifies the period; see sine and cosine transformations. The constant h does not change the amplitude or period (the shape) of the graph. It shifts the graph left (if h is negative) or right (if h is positive) and in the amount equal to h. The amount of horizontal shift is called the phase shift, which equals h.

How to determine the horizontal and vertical translations of sine and cosine?
How to graph translated sine and cosine functions? How to find the phase shift (the horizontal shift) of sine and cosine functions? How to graph a trigonometric function by graphing the original and then applying the phase shift?

### Transformations of Sine and Cosine (amplitude, period, phase shift and vertical shift)

Graph Sine and Cosine functions with the four basic transformations: amplitude, period, phase shift and vertical shift.

Graph y = Asin(B(x-D))+C and y = Acos(B(x-d))+C Example 1: Describe the transformation and graph y = −3cos(6x) Example 2: Describe the transformation and graph y = 1/2sin(x+π/4) Example 3: Describe the transformation and graph y = 2sin(1/2πx) − 1 Example 4: Describe the transformation and graph y = 4cos(4x − 8) − 1

### 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. Next, find the period of the function which is the horizontal distance for the function to repeat. If the period is more than 2pi, B is a fraction; use the formula period=2pi/B to find the exact value. Last, find any phase shift, h.
Write the equation of a sine or cosine function given a graph. Example of determining the equation of a transformed sine function from a graph. Examples of determining the equation of a transformed sine or cosine function from a graph.

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.  