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Tangent Function

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A series of free, online Trigonometry Video Lessons.
Videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

  • how to define the tangent function using the unit circle
  • how to evaluate the tangent function
  • how to graph the tangent function
  • how to transform the graph of tangent functions
  • how to find the x-intercepts and vertical asymptotes of the tangent function

The Tangent Function

In right triangle trigonometry (for acute angles only), the tangent is defined as the ratio of the opposite side to the adjacent side. The unit circle definition is tan(θ)=y/x or tan(θ)=sin(θ)/cos(θ). The tangent function is negative whenever sine or cosine, but not both, are negative: the second and fourth quadrants. Tangent is also equal to the slope of the terminal side.

How to define the tangent function for all angles?

Evaluating the Tangent Function

When evaluating the tangent function, to find values of the tangent function at different angles, we first identify the reference angle formed by the terminal side and the x-axis. Then, we find the tangent of this reference angle and, based on which quadrant the terminal side is in, decide if tangent is positive or negative. Tangent is positive in the first and third quadrants, where both sine and cosine are positive and both are negative.

How to evaluate the tangent of 3π/4?

Graph of the Tangent Function

For a tangent function graph, create a table of values and plot them on the coordinate plane. Since tan(theta)=y/x, whenever x=0 the tangent function is undefined (dividing by zero is undefined). These points, at theta=pi/2, 3pi/2 and their integer multiples, are represented on a graph by vertical asymptotes, or values the function cannot equal. Because of unit circle symmetry over the y-axis, the period is pi/2.

This video shows how to graph the Tangent function on the coordinate plane using the unit circle, how to determine the domain and range of the tangent function. How to graph the tangent function using the unit circle and reciprocal identity? How to graph the tangent function using the unit circle (animation)?

Transforming the Tangent Graph

When graphing a tangent transformation, start by using a theta and tan(theta) t-table for -pi/2 to pi/2. In the case of y = Atan(Bx) or y = Atan(B(x - h)), define Bx or B(x - h) to be equal to theta and solve for x. Now use this equation to create a x and Atan(Bx) or Atan(B(x - h)) table, which will give coordinate pairs to plot.

How to graph y = tan(x) for one or more periods?

How to graph the transformation of the basic tangent function when the period has changed?
Describe the transformation and graph y = tan(4x)

Intercepts and Asymptotes of Tangent Functions

The tangent identity is tan(theta)=sin(theta)/cos(theta), which means that whenever sin(theta)=0, tan(theta)=0, and whenever cos(theta)=0, tan(theta) is undefined (dividing by zero). When the tangent function is zero, it crosses the x-axis. Therefore, to find the intercepts, find when sin(theta)=0. To find the vertical asymptotes determine when cos(theta)=0.

How to find the x-intercepts and vertical asymptotes of the graph of y = tan(x)?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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