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

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The Graph of Cosine, y = cos (x).

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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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).

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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