In these lessons, we will look at graphing the cosine function and properties of the cosine function.

Related Topics

More lessons on Trigonometry

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 **g****raph of y = cos **

** **

**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 |
1 |
0 |
–1 |
0 |
1 |

2 cos |
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˚.

Graphing basic sine and cosine functions (in degrees).

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