Introduction to the Tangent Function

So far in Unit 3, we have spent a lot of time with sine and cosine. They describe the \(x\) and \(y\) coordinates of a point on the unit circle. But what happens when we look at the ratio of those coordinates? That is where the tangent function comes in! While sine and cosine create smooth, continuous waves, the tangent function introduces some exciting "breaks" in the graph called vertical asymptotes. In this chapter, we will explore how the tangent function behaves, what its graph looks like, and why its period is different from the functions we’ve seen before.

1. Defining Tangent

In Chapter 3.2, we learned that for any angle \( \theta \), the tangent is defined as the ratio of sine to cosine. Analytically, we write this as:

\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

Because sine represents the vertical change (\(y\)) and cosine represents the horizontal change (\(x\)) on the unit circle, you can think of tangent as the slope of the terminal ray of the angle \( \theta \). Since slope is "rise over run," \( \tan(\theta) = \frac{y}{x} \).

Did you know? The word "tangent" comes from the Latin word tangere, meaning "to touch." In geometry, a tangent line touches a circle at exactly one point!


2. Domain, Range, and Vertical Asymptotes

Unlike sine and cosine, which are defined for every possible angle, the tangent function has a few "forbidden zones."

The Domain and Vertical Asymptotes

Since \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), the function is undefined whenever the denominator is zero. This happens when \( \cos(\theta) = 0 \). On the unit circle, the cosine is zero at the "top" and "bottom," which corresponds to angles like \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), \( -\frac{\pi}{2} \), and so on.

At these inputs, the graph of the tangent function has vertical asymptotes. These are vertical lines that the graph approaches but never touches or crosses. The equations for these asymptotes are:

\( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

The Range

As the angle \( \theta \) gets closer to \( \frac{\pi}{2} \), the value of \( \cos(\theta) \) gets very small, which makes the fraction \( \frac{\sin(\theta)}{\cos(\theta)} \) get incredibly large. Because of this, the range of the tangent function is all real numbers, or \( (-\infty, \infty) \).

Key Takeaway: The domain is all real numbers except where \( \cos(\theta) = 0 \). The range has no boundaries—it goes from negative infinity to positive infinity!


3. The Period of Tangent

This is a common "gotcha" for students! While the period of sine and cosine is \( 2\pi \), the period of the tangent function is \( \pi \).

Why? Because the slope of a line repeats every \( 180^\circ \) (or \( \pi \) radians). If you have a line with a certain slope and you rotate it half a circle, it points in the opposite direction but still lies on the same line, so it has the same slope. Therefore, the tangent values repeat every \( \pi \) units.

Quick Formula: If you have a function \( f(x) = \tan(bx) \), the period is calculated as:

\( \text{Period} = \frac{\pi}{|b|} \)


4. Graphing the Tangent Function

To sketch one cycle of the parent function \( f(\theta) = \tan(\theta) \), it is easiest to look at the interval between two asymptotes, such as \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

Here are the five key features to plot for one cycle:

  • Vertical Asymptote: \( \theta = -\frac{\pi}{2} \)
  • Low Point: At \( \theta = -\frac{\pi}{4} \), \( \tan(\theta) = -1 \)
  • Center Point (Zero): At \( \theta = 0 \), \( \tan(\theta) = 0 \)
  • High Point: At \( \theta = \frac{\pi}{4} \), \( \tan(\theta) = 1 \)
  • Vertical Asymptote: \( \theta = \frac{\pi}{2} \)

The graph looks like a series of repeating "S-curves" that increase from left to right between the asymptotes. This means the tangent function is increasing on every interval in its domain.


5. Transformations of the Tangent Function

Just like the sinusoidal functions we studied in Chapter 3.6, we can transform the tangent function using the general form:

\( f(x) = a \tan(b(x - c)) + d \)

Vertical Dilation (\( a \)): Tangent does not have an "amplitude" because it goes to infinity. However, \( |a| \) acts as a vertical stretch or compression. If \( a \) is negative, the graph is reflected over the horizontal midline.

Horizontal Dilation (\( b \)): This affects the period. Remember, \( \text{Period} = \frac{\pi}{b} \).

Phase Shift (\( c \)): This shifts the graph left or right.

Vertical Shift (\( d \)): This shifts the graph up or down. The "center point" (which is usually at \( (0,0) \)) moves to \( (c, d) \).


Summary and Common Mistakes

Quick Review Box:
- Ratio: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
- Period: \( \pi \) (not \( 2\pi \)!)
- Asymptotes: Located where \( \cos(x) = 0 \).
- Range: \( (-\infty, \infty) \)

Common Mistakes to Avoid:
1. Using the wrong period: Don't use \( \frac{2\pi}{b} \) for tangent; always use \( \frac{\pi}{b} \).
2. Mixing up sine and cosine: Remember that tangent is \( \frac{y}{x} \). If you flip them, you are actually finding the cotangent (which we will learn in Chapter 3.11).
3. Forgetting asymptotes: When asked for the domain, always remember to exclude the values where the function "blows up" to infinity.

Don't worry if the graph looks strange at first! Unlike the smooth waves of sine, tangent is all about those sharp breaks at the asymptotes. Practice identifying where \( \cos(x) = 0 \), and the rest of the graph will fall into place.