Cosine Graph

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

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

y = cos θ is known as the cosine function.

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

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

Properties of the cosine function:

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

Example:

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

Solution:
We know that cos 180˚ = –1. So, p = –1.

We know that for a cosine graph, cos θ = 0 for θ = 90˚ and 270˚.  So, θ = 90˚

We know that for a cosine graph, cos θ = 1 for θ = 0˚ and 360˚. 
So, r = 360˚

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

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

Plot the points and join with a smooth curve.

Also take note that, the graphs of y = sin x and y = cos x, for 0˚ ≤ x ≤ 360˚, intersect at two points: x = 45˚ and x = 225˚.

Videos

Graphing basic sine and cosine functions (in degrees)

An introduction to the graphs of sine and cosine functions (in radians)

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Trigonometry