Tangent Graph

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 y against the corresponding values of θ.
The graph of y = tan θ, for 0˚ ≤ θ ≤ 360˚ obtained is as shown:

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 θdoes not have any maximum or minimum values. The range of values of tan θis –∞ < 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˚.

Example:

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

Solution:
Set up a table of values for y = tan x

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˚.

Videos

Graphing basic tangent functions

Graphing the Tangent, Secant, Cosecant, and Cotangent Functions Professor Edward Burger explains graphing the tangent, secant, cosecant, and cotangent functions

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