The Tangent Function - AP PreCalculus Study Guide

The Core Idea: The Tangent Function

The tangent function provides a way to describe periodic phenomena that feature vertical asymptotes, or points of infinite discontinuity. Unlike the sine and cosine functions which model smooth, continuous waves, the tangent function is defined as the ratio of the sine and cosine functions. This fundamental definition, $tan(x)=sin(x)/cos(x)$, is the key to all of its unique properties.

Because the tangent function is a ratio, its behavior is dictated by the values of its numerator ($sin(x)$) and its denominator ($cos(x)$). Where the denominator, $cos(x)$, is zero, the function is undefined, resulting in vertical asymptotes. Where the numerator, $sin(x)$, is zero, the entire function is zero, resulting in x-intercepts. This interplay creates a periodic function with a repeating pattern of curves separated by asymptotes, making it suitable for modeling phenomena with cyclical, abrupt changes. The study of the tangent function involves understanding how its graphical and analytical properties—such as its domain, range, period, and asymptotes—are all direct consequences of its definition as a ratio.

Key Formulas & Definitions

The analytical properties of the tangent function are derived from its definition and the parameters of its general form, $f(x)=a\tan (b(x-c))+d$.

Foundational Definition

The tangent function is defined as the ratio of the sine function to the cosine function.

$$\tan (x)=\frac{\sin (x)}{\cos (x)}$$

Period

The period is the length of the smallest horizontal interval over which the function's values repeat.

• Parent Function: The period of $f(x)=\tan (x)$ is $\pi$.

• Transformed Function: The period $P$ of $f(x)=a\tan (b(x-c))+d$ is calculated using the parameter $b$.

  $$P=\frac{\pi }{|b|}$$

Vertical Asymptotes

Vertical asymptotes are vertical lines that the graph of the function approaches but never touches. They occur where the function is undefined.

• Condition: Vertical asymptotes exist where the denominator of the tangent function, $\cos (x)$, is equal to zero.

• Parent Function: For $f(x)=\tan (x)$, vertical asymptotes occur at:

  $$x=\frac{\pi }{2}+k\pi ,\text{for any integer }k$$

• Transformed Function: For $f(x)=a\tan (b(x-c))+d$, find the asymptotes by setting the argument of the function equal to the locations of the parent function's asymptotes:

  $$b(x-c)=\frac{\pi }{2}+k\pi$$

X-Intercepts

X-intercepts are the points where the graph of the function crosses the x-axis.

• Condition: X-intercepts exist where the numerator of the tangent function, $\sin (x)$, is equal to zero (and the denominator is not).

• Parent Function: For $f(x)=\tan (x)$, x-intercepts occur where $\sin (x)=0$.

Understanding the Ratio-Based Properties

The most critical nuance of the tangent function is that all of its primary characteristics—domain, range, asymptotes, and intercepts—are direct consequences of its definition as $\tan (x)=\sin (x)/\cos (x)$.

• Domain and Asymptotes: A rational expression is undefined when its denominator is zero. For the tangent function, this means the function is undefined whenever $\cos (x)=0$. The values of $x$ for which $\cos (x)=0$ are $x=\dots ,-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\dots$, which can be generalized as $x=\frac{\pi }{2}+k\pi$ for any integer $k$. These are the exact locations of the vertical asymptotes. Therefore, the domain of the tangent function is the set of all real numbers except for these values.

• X-Intercepts: A fraction is equal to zero only when its numerator is zero and its denominator is non-zero. For the tangent function, this means $\tan (x)=0$ whenever $\sin (x)=0$. The values of $x$ for which $\sin (x)=0$ are $x=\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots$, which can be generalized as $x=k\pi$ for any integer $k$. These are the locations of the x-intercepts.

• Range: As $x$ approaches a value where $\cos (x)$ is zero (an asymptote), the value of $\cos (x)$ becomes an infinitesimally small number. Dividing $\sin (x)$ (which is close to 1 or -1 near these points) by a very small number results in a very large number. This causes the function's output to approach positive or negative infinity. Consequently, the tangent function can take on any real number value, and its range is $(-\infty ,\infty )$.

Core Concepts & Rules

• Fundamental Identity: The tangent function is defined by the identity $\tan (x)=\sin (x)/\cos (x)$.

• Periodicity: The tangent function is periodic with a fundamental period of $\pi$.

• Domain: The domain of $f(x)=\tan (x)$ is the set of all real numbers $x$ such that $x\ne \frac{\pi }{2}+k\pi$ for any integer $k$.

• Range: The range of $f(x)=\tan (x)$ is the set of all real numbers, $(-\infty ,\infty )$.

• Vertical Asymptotes: The graph of $f(x)=\tan (x)$ has vertical asymptotes at the values excluded from its domain, $x=\frac{\pi }{2}+k\pi$.

• X-Intercepts: The graph of $f(x)=\tan (x)$ has x-intercepts where $\sin (x)=0$.

• Transformations: The function $f(x)=a\tan (b(x-c))+d$ represents a transformation of the parent tangent function.

  • $a$: A vertical stretch or compression by a factor of $|a|$. If $a<0$, the function is reflected across the horizontal midline.

  • $b$: A horizontal stretch or compression. The new period is $P=\pi /|b|$.

  • $c$: A horizontal translation (phase shift) of $c$ units.

  • $d$: A vertical translation of $d$ units. The line $y=d$ becomes the new midline of the function.

Step-by-Step Example 1: Analyzing the Parent Function

Problem: Analyze the function $f(x)=\tan (x)$. Determine its period, domain, range, the locations of its vertical asymptotes, and its x-intercepts.

Step 1: Start with the Fundamental Definition

The function is $f(x)=\tan (x)$, which is defined as $\tan (x)=\frac{\sin (x)}{\cos (x)}$.

Step 2: Determine the Period

Based on the essential knowledge for the tangent function, the fundamental period is $\pi$. This means the function's pattern repeats every $\pi$ units.

Step 3: Find the Vertical Asymptotes and Domain

Vertical asymptotes occur where the denominator, $\cos (x)$, is zero.

• Set the denominator to zero: $\cos (x)=0$.

• Recall from the unit circle that $\cos (x)=0$ at $x=\frac{\pi }{2}$, $x=\frac{3\pi }{2}$, and so on.

• Generalize this pattern: The asymptotes are located at $x=\frac{\pi }{2}+k\pi$ for any integer $k$.

• The domain is all real numbers except for the values where the asymptotes occur.

• Domain:$\{x\in \mathbb{R}\mid x\ne \frac{\pi }{2}+k\pi ,k\in \mathbb{Z}\}$.

Step 4: Find the X-Intercepts

X-intercepts occur where the numerator, $\sin (x)$, is zero.

• Set the numerator to zero: $\sin (x)=0$.

• Recall from the unit circle that $\sin (x)=0$ at $x=0$, $x=\pi$, $x=2\pi$, and so on.

• Generalize this pattern: The x-intercepts are located at $x=k\pi$ for any integer $k$.

Step 5: Determine the Range

The tangent function is unbounded as it approaches its vertical asymptotes.

• Range:$(-\infty ,\infty )$.

Summary of Properties for $f(x)=\tan (x)$:

• Period:$\pi$

• Domain: All real numbers except $x=\frac{\pi }{2}+k\pi$

• Range: All real numbers

• Vertical Asymptotes:$x=\frac{\pi }{2}+k\pi$

• X-Intercepts:$x=k\pi$

Step-by-Step Example 2: Analyzing a Transformed Tangent Function

Problem: Consider the function $g(x)=-4\tan (\frac{1}{2}(x+\frac{\pi }{3}))$. Identify the transformations applied to $f(x)=\tan (x)$ to create $g(x)$. Then, determine the period and the equations of two consecutive vertical asymptotes of $g(x)$.

Step 1: Identify the Transformation Parameters

Compare $g(x)$ to the general form $f(x)=a\tan (b(x-c))+d$.

• $a=-4$

• $b=\frac{1}{2}$

• $c=-\frac{\pi }{3}$ (since the form is $x-c$, $x+\frac{\pi }{3}$ means $c$ is negative)

• $d=0$

Step 2: Describe the Transformations

• $a=-4$: A vertical stretch by a factor of 4 and a reflection across the x-axis.

• $b=\frac{1}{2}$: A horizontal stretch by a factor of $1/|b|=1/(1/2)=2$.

• $c=-\frac{\pi }{3}$: A horizontal translation (phase shift) of $\frac{\pi }{3}$ units to the left.

• $d=0$: No vertical translation.

Step 3: Calculate the Period

Use the formula $P=\frac{\pi }{|b|}$.

• $P=\frac{\pi }{|\frac{1}{2}|}=\frac{\pi }{\frac{1}{2}}=2\pi$.

• The period of $g(x)$ is $2\pi$.

Step 4: Find the Equations of Consecutive Vertical Asymptotes

The asymptotes of the parent function $\tan (\theta )$ occur when $\theta =\frac{\pi }{2}+k\pi$. For our function $g(x)$, the argument is $\theta =\frac{1}{2}(x+\frac{\pi }{3})$.

• Set the argument equal to the locations of two consecutive parent asymptotes, for example, at $k=0$ and $k=1$.

  • Asymptote 1 (for $k=0$): $\frac{1}{2}(x+\frac{\pi }{3})=\frac{\pi }{2}$

  • Asymptote 2 (for $k=1$): $\frac{1}{2}(x+\frac{\pi }{3})=\frac{3\pi }{2}$

• Solve for $x$ for the first asymptote:

  $$\frac{1}{2}\left(x+\frac{\pi }{3}\right)=\frac{\pi }{2}$$

  $$x+\frac{\pi }{3}=\pi$$

  $$x=\pi -\frac{\pi }{3}=\frac{2\pi }{3}$$

• Solve for $x$ for the second asymptote:

  $$\frac{1}{2}\left(x+\frac{\pi }{3}\right)=\frac{3\pi }{2}$$

  $$x+\frac{\pi }{3}=3\pi$$

  $$x=3\pi -\frac{\pi }{3}=\frac{8\pi }{3}$$

• Two consecutive asymptotes are at $x=\frac{2\pi }{3}$ and $x=\frac{8\pi }{3}$.

• Verification: The distance between these asymptotes is $\frac{8\pi }{3}-\frac{2\pi }{3}=\frac{6\pi }{3}=2\pi$, which matches the period calculated in Step 3.

Using Your Calculator

A graphing calculator is an excellent tool for visualizing the tangent function and verifying its analytical properties.

Task: Graph the function $g(x)=-4\tan (\frac{1}{2}(x+\frac{\pi }{3}))$ from Example 2 to verify its period and asymptotes.

Step 1: Set Mode

• Press the [MODE] button.

• Ensure that RADIAN is selected, not `DEGREE$. The properties of $\tan (x)$ are defined in terms of $\pi$ radians.

Step 2: Enter the Function

• Press the [Y=]` button. - In `Y1`, type the function. Be careful with parentheses: `Y1 = -4 * tan( (1/2) * (X + π/3) )`. **Step 3: Set the Viewing Window** - Press the `[WINDOW]` button. This is the most critical step for tangent. - We calculated the period to be $2\pi. To see at least one full cycle clearly, set the x-axis limits around the calculated asymptotes.

• Our asymptotes were at $2\pi /3\approx 2.1$ and $8\pi /3\approx 8.4$.

• Set $Xmin=0$.

• Set $Xmax=3\pi$ (which is approximately 9.4, showing just past the second asymptote).

• Set $Xscl=\pi /3$ to have tick marks at helpful intervals.

• The range of tangent is all real numbers, so $Ymin$ and $Ymax$ can be set to capture the shape near the midline. $Ymin=-10$ and $Ymax=10$ is a good starting point.

Step 4: Graph and Analyze

• Press the [GRAPH] button.

• You should see the characteristic shape of the tangent curve. Because $a=-4$ is negative, the curve should decrease from left to right between the asymptotes.

• The calculator may attempt to draw a near-vertical line at the asymptotes. This is a graphing artifact; it is not part of the function. You can visually confirm that the graph "blows up" near $x=2\pi /3$ and $x=8\pi /3$.

• You can also see that the graph completes one full cycle between these two asymptotes, visually confirming the period is $2\pi$.

AP Exam Quick Hit

Common Question Types

• Identifying Properties from an Equation: Given a function like $f(x)=5\tan (4x)-1$, you may be asked to state its period.

  • Example: What is the period of $f(x)=5\tan (4x)-1$? (Answer: $P=\pi /|4|=\pi /4$).

• Finding Asymptotes from an Equation: You will be asked to find the equation of one or all vertical asymptotes for a given tangent function.

  • Example: Find the equation for one vertical asymptote of $g(x)=\tan (x-\pi /4)$. (Answer: Set $x-\pi /4=\pi /2$, so $x=3\pi /4$).

• Determining an Equation from a Graph: A graph of a transformed tangent function is provided, and you must determine the values of the parameters $a$, $b$, $c$, or $d$.

  • Example: The graph shown has consecutive asymptotes at $x=0$ and $x=2$. What is the value of $b$ in its equation $y=\tan (bx)$? (Answer: The period is `2-0=2$. So $2=\pi /|b|$, which means $|b|=\pi /2$).

Common Mistakes

• Incorrect Period Formula: Using the period formula for sine/cosine ($2\pi /|b|$) instead of the correct formula for tangent ($\pi /|b|$). The period of the parent tangent function is $\pi$, not $2\pi$.

• Asymptote Calculation Errors: When finding asymptotes for $\tan (b(x-c))$, students often just shift the parent asymptote by $c$ instead of solving the full equation $b(x-c)=\pi /2+k\pi$. The horizontal dilation ($b$) must be accounted for first.

• Confusing Intercepts and Asymptotes: Mixing up the conditions for intercepts and asymptotes. Remember: Asymptotes come from the denominator ($\cos (x)=0$), and x-intercepts come from the numerator ($\sin (x)=0$).

• Domain and Range Errors: Incorrectly stating that the domain is all real numbers or that the range is restricted like sine or cosine. The tangent function's domain has infinite exclusions, and its range is all real numbers.