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Sine Graph




 
In this lesson, we will look at how to graph the sine function, the properties of the sine function and the unit circle definition of the sine function.

Related Topics: More Trigonometry Lessons

We will start with the unit circle.
A unit circle is a circle of radius one unit with its center at the origin.

unit circle

For a point (x, y) on the unit circle;

y = sin θ is known as the sine function.

Using the unit circle, we can plot the values of y against the corresponding values of θ.

The graph of y = sin θ, for 0˚ ≤ θ ≤ 360˚ obtained is as shown:

unit circle for sine



Properties of the sine function:

  • The sine function forms a wave that starts from the origin
  • sin θ= 0 when θ = 0˚, 180˚, 360˚.
  • Maximum value of sin θ is 1 when θ= 90˚. Minimum value of sin θ is –1 when θ = 270˚. So, the range of values of sin θ is
  • –1 ≤ sin θ ≤ 1
  • As the point P moves round the circle in either the clockwise or anticlockwise direction, the sine curve above repeats itself for every interval of 360˚. The interval over which the sine wave repeats itself is called the period.
sine graph

Example:
The diagram shows a graph of  y = sin x for 0˚ ≤ ≤ 360˚, determine the values of p, q and r.

Solution:
In order to answer this type of questions, you will need to remember the general properties (or shape) of a sine graph.

We know that the maximum value of a sine graph is 1. So, p = 1.
We know that for a sine graph, sin θ = 0 for θ= 0˚, 180˚ and 360˚. So, θ= 180˚
We know that for a sine graph, sin θ = –1 for θ= 270˚. So, r = 270˚

Example:
Sketch the graph of y = sin 2x for 0˚ ≤ 2x ≤ 360˚.

Solution:
Set up a table of values for the equation y = sin 2x

x

0

45

90

135

180

2x

0

90

180

270

360

sin 2x

0

1

0

–1

0


Plot the points and join with a smooth curve.



 
Graphing basic sine and cosine functions (in degrees)
How to use the unit circle definition of the sine function to make a graph of it? (in radians)


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