In these lessons, we will look at graphing the tangent function.

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More Trigonometry Lessons

**Example:**

Sketch the graph of*y* = tan *x* for 0˚ ≤ *x *≤ 360˚.

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Graphing the Tangent Function

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Related Topics:

More Trigonometry Lessons

We will start with the unit circle.

A **unit circle** is a circle of radius one unit with its centre at the origin.

In the following diagram, the line *l* is a tangent to the unit circle at point *P*.

We can see that:

** y = tan θ is known as the tangent function**. Using the unit circle, we can plot the values of

The

**Properties of the tangent function:**

- The curve is not continuous. It breaks at
**θ***=*90˚ and 270˚, where the function is undefined - tan
**θ**= 0 when**θ**= 0˚, 180˚, 360˚. tan**θ**= 1 when**θ**= 45˚ and 225˚. - tan
**θ**= –1 when**θ**= 135˚ and 315˚. - tan
**θ**does not have any maximum or minimum values. The range of values of tan**θ**is –∞ < tan**θ**< ∞ . - As the point
*P*moves round the circle in either the clockwise or anticlockwise direction, the tangent curve above repeats itself for every interval of**180˚**. Its period is 180˚.

Sketch the graph of

**Solution:**

Set up a table of values for *y* = tan *x*

x |
0 |
45 |
90 |
135 |
180 |
225 |
270 |
315 |
360 |

tan |
0 |
1 |
undefined |
–1 |
0 |
1 |
undefined |
–1 |
0 |

Plot the points and join with a smooth curve.

**Example:**

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

*Solution:*

We know that for a tangent graph, tan** θ** = 1 when **θ**= 45˚ and 225˚. So, *b *= 45˚.

We know that for a tangent graph, tan** θ** = 0 when** θ**= 0˚, 180˚ and 360˚. So, *c* = 180˚.

Graphing the Tangent Function.

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