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 center at the origin.
The following diagram shows the unit circle and the cosine graph. Scroll down the page for more examples and solutions on how to graph the cosine function.
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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˚ (2π).
Peaks: At x = 0, 2π, 4π,… (cos x = 1).
Troughs: At x = π, 3π,… (cos x = -1).
Zeros: At \(x = \frac{π}{2}, \frac{3π}{2},\) ,… (cos x = 0).
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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