# How do you graph two cycles of #y=2tan(3theta)#?

##### 1 Answer

#### Answer:

Try sketching with references to certain properties of the graph, noticeably intersects, monotonies, and asymptotes.

graph{tan(3x) [-2.23, 2.23, -1.162, 1.162]}

#### Explanation:

Note that all

**1. Intersections**

Evaluate the function at

We can find

Let

This explanation shows how to solve the equation by considering the composite nature of the function: it consists of two parts, an inner function

Substituting

Thus coordinates of

Taking

**2. Monotonies**

The tangent function always increases as the angle grows, as seen from a unit circle. Therefore the graph of

**3.Asymptotes**

The tangent function is not defined at the sum of

For the composite tangent function here, the general expression for all the asymptotes would be

Evaluating at the expression at

Now plot all three of these features on the graph, and the curve you sketch should:

a. Slopes upwards;

b. Passes through all of the intersections, and

c. Approaches, but never touches each of the asymptotes.

See also:

https://www.mathsisfun.com/geometry/unit-circle.html

Value of the tangent function on a unit circle