Cosine Graph
More lessons on Trigonometry
In these lessons, we will look at graphing the cosine function and properties of the cosine function.
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
| x | 0 |
90 |
180 |
270 |
360 |
cos x |
1 |
0 |
–1 |
0 |
1 |
2 cos x |
2 |
0 |
–2 |
0 |
2 |
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˚.
How to graph basic sine and cosine functions (in degrees)?
The Graph of Cosine, y = cos (x)
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