Table of Contents

Introduction
Understanding the Tangent Function and Its Key Properties
- Properties of the Tangent Function
Period and Vertical Asymptotes of a Tangent Function
Some Problems
Finding Exact Values – Step-by-Step Example
FAQs : Frequently Asked Questions

3.8 The Tangent Function in AP Precalculus

Utkarsh
March 3, 2025

Table of Contents

Introduction
Understanding the Tangent Function and Its Key Properties
- Properties of the Tangent Function
Period and Vertical Asymptotes of a Tangent Function
Some Problems
Finding Exact Values – Step-by-Step Example
FAQs : Frequently Asked Questions

Introduction

The tangent function is one of the fundamental trigonometric functions, playing a crucial role in AP Precalculus. Unlike sine and cosine, the tangent function has vertical asymptotes, making its graph distinct and requiring special attention when analyzing its properties. Understanding how to determine its period, vertical asymptotes, and transformations is essential for solving problems related to tangent functions.

In this blog, we will explore the key properties of the tangent function, how to find its period and vertical asymptotes, and how to write equations from graphs. We will also discuss methods for solving equations involving tangent functions and highlight common mistakes students make. By the end, you’ll have a strong grasp of how to analyze and work with tangent functions effectively in AP Precalculus. We have enlisted Top 10 books for AP Precalculus to help you in finding the effective resources easily.

Understanding the Tangent Function and Its Key Properties

The tangent function, denoted as $\tan(x)$ , is a fundamental trigonometric function that describes the ratio of the sine and cosine of an angle:

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

Since it depends on cosine, the tangent function is undefined whenever $\cos(x) = 0$ . This results in vertical asymptotes at certain points, giving the tangent function a unique behavior compared to sine and cosine.

Properties of the Tangent Function

Domain:
- The tangent function is undefined where $\cos(x) = 0$ , which occurs at: $x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$
- This means the domain is all real numbers except at these asymptotes.
Range:
- The function extends infinitely in both the positive and negative directions: $(-\infty, \infty)$
- Unlike sine and cosine, which are bounded between -1 and 1, the tangent function is unbounded.
Period:
- The standard tangent function repeats every $\pi$ units, meaning: $\tan(x + \pi) = \tan(x)$
- If the function is transformed, the period changes based on the B value in the equation $f(x) = A \tan(Bx + C) + D$ .
Vertical Asymptotes:
- Since tangent is undefined at $x = \frac{\pi}{2} + k\pi$ , vertical asymptotes occur at these points.
- When transformed, the general formula for vertical asymptotes is: $x = \frac{\pi}{2B} + k\frac{\pi}{B} - \frac{C}{B}, \quad k \in \mathbb{Z}$
Intercepts:
- The function crosses the x-axis at integer multiples of $\pi$ : $x = k\pi, \quad k \in \mathbb{Z}$
- These are the zeroes of the tangent function.
Graphical Behavior:
- The tangent graph consists of repeating S-shaped curves, with each cycle separated by vertical asymptotes.
- The function increases from $-\infty$ to $\infty$ in each cycle.
- It has point symmetry about the origin, making it an odd function: $\tan(-x) = -\tan(x)$

Period and Vertical Asymptotes of a Tangent Function

The tangent function plays a crucial role in trigonometry, particularly in modeling periodic phenomena. Unlike sine and cosine, which oscillate between fixed values, the tangent function has asymptotes where it is undefined. Understanding its period and vertical asymptotes helps in solving equations and graphing transformations efficiently.

1. Standard Tangent Function

The parent function for tangent is:

$y = \tan x$

This function has a period of $\pi$ , meaning it repeats itself every $\pi$ units along the x-axis. The standard vertical asymptotes occur at:

$x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$

where $k$ represents any integer.

The function exhibits the following properties:

It increases from negative to positive infinity within each cycle.
It has vertical asymptotes where the function is undefined.
It has x-intercepts at $x = k\pi$ , where $k$ is an integer.

2. General Form of a Tangent Function

When the tangent function is transformed, it takes the form:

$y = A \tan(Bx + C) + D$

where:

$A$ affects the vertical stretch (amplitude does not apply to tangent).
$B$ affects the horizontal stretch or compression, which determines the new period.
$C$ shifts the function horizontally (phase shift).
$D$ shifts the function vertically.

The period of the tangent function is modified by $B$ and is given by:

$\text{New Period} = \frac{\pi}{|B|}$

3. Finding the Period of a Tangent Function

To determine the period of $y = \tan(Bx)$ :

Identify the coefficient $B$ in the equation.
Use the formula: $\text{Period} = \frac{\pi}{|B|}$
If $B > 1$ , the function compresses horizontally (reduces the period).
If $0 < B < 1$ , the function stretches horizontally (increases the period).

Example 1:

Find the period of $y = \tan(2x)$ .

Here, $B = 2$ .
Using the formula: $\text{Period} = \frac{\pi}{|2|} = \frac{\pi}{2}$
The function repeats every $\frac{\pi}{2}$ instead of $\pi$ , meaning the graph is compressed.

Example 2:

Find the period of $y = \tan\left(\frac{1}{2}x\right)$ .

Here, $B = \frac{1}{2}$ .
Using the formula: $\text{Period} = \frac{\pi}{|1/2|} = 2\pi$
The function repeats every $2\pi$ , meaning it is stretched horizontally.

4. Finding the Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, which happens when the denominator of the tangent function equals zero.

For the standard tangent function $y = \tan x$ , the vertical asymptotes are at:

$x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$

For a transformed tangent function $y = \tan(Bx + C)$ , the vertical asymptotes occur when:

$Bx + C = \frac{\pi}{2} + k\pi$

To solve for $x$ :

Isolate $x$ : $x = \frac{\pi}{2B} - \frac{C}{B} + \frac{k\pi}{B}$
This gives the general formula for the asymptotes.

Example 3:

Find the vertical asymptotes of $y = \tan(2x)$ .

The equation is $2x = \frac{\pi}{2} + k\pi$ .
The vertical asymptotes occur at $x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \dots$ .
The vertical asymptotes occur at $x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \dots$ .

Example 4:

Find the vertical asymptotes of $y = \tan\left(\frac{1}{2}x\right)$ .

The equation is $\frac{1}{2}x = \frac{\pi}{2} + k\pi$ .
Solving for $x$ : $x = \pi + 2k\pi$
The vertical asymptotes occur at $x = \pi, 3\pi, 5\pi, \dots$ .

Download this free AP Precalculus worksheet with solutions for targeted practice and a better understanding of concepts

Start preparing effectively now!

Some Problems

Step 1: Recall the Standard Period of $<span class="katex">y = \tan x$

The parent function of the tangent function is: $y = \tan x$
The period of the standard tangent function is $\pi$ .
This means the function repeats itself every $\pi$ units along the x-axis.

Step 2: Identify the Vertical Asymptotes

The standard tangent function has vertical asymptotes at: $x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$
In this graph, the vertical asymptotes are at: $x = -2\pi, -\pi, 0, \pi, 2\pi$
Observing these values, we see that the distance between consecutive asymptotes is: $\pi - (-\pi) = 2\pi$
This suggests that the function repeats every $2\pi$ instead of $\pi$ .

Step 3: Compare with the Standard Tangent Function

The general form of the transformed tangent function is: $y = \tan(Bx)$
The period of a transformed tangent function is given by: $\frac{\pi}{B}$
From our observation, the period of the given function is $2\pi$ . $\frac{\pi}{B} = 2\pi$
Solving for $B$ : $B = \frac{1}{2}$
This means the function has been horizontally stretched, which increased its period from $\pi$ to $2\pi$ .

Step 4: Conclusion

The period of the given function $f(x)$ is $2\pi$ .
This confirms that the function is a horizontally stretched version of $y = \tan x$ , with $B = \frac{1}{2}$ .

Final Answer:

$\mathbf{2\pi}$

Finding Exact Values – Step-by-Step Example

Example 1: Find the exact value of sin(150°).

Step 1: Identify the reference angle.

150° is in Quadrant II.
Reference angle = 180° – 150° = 30°.

Step 2: Determine the sign.

Since sine is positive in Quadrant II, we use sin(30°) = 1/2.

Answer: sin(150°) = 1/2

Example 2: Find the exact value of tan(300°).

Step 1: Find the reference angle.

300° is in Quadrant IV.
Reference angle = 360° – 300° = 60°.

Step 2: Determine the sign.

Since tangent is negative in Quadrant IV, we use tan(60°) = √3.

Answer: tan(300°) = -√3

Example 3: Find the exact value of cos(7π/6).

Step 1: Convert to degrees.

7π/6 = 210°

Step 2: Identify the reference angle.

210° is in Quadrant III.
Reference angle = 210° – 180° = 30°.

Step 3: Determine the sign.

Cosine is negative in Quadrant III.
cos(30°) = √3/2, so cos(7π/6) = -√3/2.

FAQs : Frequently Asked Questions

Q-1 : How to Study for AP Precalculus?

AP Precalculus is a new course designed to strengthen students’ understanding of functions, modeling, and analytical problem-solving. To study effectively:

Understand the Format – Familiarize yourself with the topics covered, such as polynomial, rational, exponential, logarithmic, and trigonometric functions. The exam also tests real-world modeling and function transformations.
Use the Official Course and Exam Description (CED) – The College Board provides a CED that outlines all the concepts tested. Make sure to review it carefully.
Build a Strong Algebraic Foundation – Since AP Precalculus builds on Algebra 2, ensure you’re comfortable with algebraic manipulations, inequalities, and solving different types of equations.
Practice with AP-Style Questions – Work through multiple-choice and free-response questions from AP resources, review books, and online platforms like AP Classroom.
Master Graphing Techniques – Graphing functions, transformations, asymptotes, and intersections are essential. Practice using a graphing calculator effectively.
Solve Real-World Problems – The course emphasizes modeling, so work on applying functions to real-life situations like exponential growth, trigonometric modeling, and finance-related problems.
Review Previous Mistakes – Regularly analyze your errors in practice tests to avoid repeating them.
Use Flashcards for Key Concepts – Memorize important formulas, function behaviors, and trigonometric identities.
Take Timed Practice Tests – Simulate real test conditions to improve time management and endurance.

Q-2 : How Long Is the AP Precalculus Exam?

The AP Precalculus exam is 2 hours and 14 minutes long and consists of two sections:

Section 1: Multiple Choice (40 Questions, 2 hours, 62% of the score)
- Part A: 28 questions (No calculator)
- Part B: 12 questions (Calculator allowed)
Section 2: Free-Response (4 Questions, 1 hour, 37.5% of the score)
- Part A: 2 questions; 30 minutes; 18.75% of exam score (graphing calculator required)
- Part B: 2 questions; 30 minutes; 18.75% of exam score (calculator not permitted)

Need personalized guidance? Book a private online AP Precalculus tutor now!

Keep an eye on the official College Board web page for current information regarding the Exam!

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