What is the Graphical Representation of tan x?

S6-SA2-0399

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The graphical representation of tan x is a curve that shows how the value of tan x changes as the angle x changes. It's a special type of graph with repeating patterns and vertical lines where the function is undefined, meaning it goes to 'infinity'.

Simple Example

Quick Example

Imagine you are watching a swing move. The height of the swing changes over time, sometimes going up, sometimes down. If you plot this height against time, you get a wavy pattern. Similarly, the graph of tan x shows how its value 'swings' but in a more complex, repeating way, especially as the angle changes from 0 to 90 degrees, then 90 to 180 degrees, and so on.

Worked Example

Step-by-Step

Let's plot some points for y = tan x:

1. For x = 0 degrees, tan(0) = 0. So, (0, 0) is a point.
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2. For x = 30 degrees, tan(30) = 1/sqrt(3) which is approximately 0.577. So, (30, 0.577) is a point.
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3. For x = 45 degrees, tan(45) = 1. So, (45, 1) is a point.
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4. For x = 60 degrees, tan(60) = sqrt(3) which is approximately 1.732. So, (60, 1.732) is a point.
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5. As x gets closer to 90 degrees (e.g., 89 degrees), tan x becomes very large. Tan(89) is about 57.29.
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6. At x = 90 degrees, tan x is undefined. This means the graph will have a vertical line (called an asymptote) at x = 90 degrees, which the curve approaches but never touches.
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7. For x = 120 degrees, tan(120) = -sqrt(3) which is approximately -1.732. So, (120, -1.732) is a point.
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8. For x = 180 degrees, tan(180) = 0. So, (180, 0) is a point.

If you plot these points and connect them, you'll see the 'S'-shaped repeating pattern of the tan x graph, with vertical lines at 90, 270 degrees, and so on.

Why It Matters

Understanding the tan x graph is crucial in fields like Physics for analyzing wave patterns and oscillations, and in Engineering for designing circuits or understanding signal processing. It's also used in Computer Graphics to create realistic 3D models and animations. Careers in AI/ML often use such periodic functions for pattern recognition.

Common Mistakes

MISTAKE: Thinking the tan x graph is continuous everywhere, like a straight line or a simple curve. | CORRECTION: Remember that tan x is undefined at 90 degrees, 270 degrees, and so on. These points have vertical asymptotes, making the graph discontinuous at those specific angles.

MISTAKE: Forgetting that the graph of tan x repeats itself every 180 degrees (or pi radians). | CORRECTION: The tan x function has a period of 180 degrees. This means the pattern you see between -90 and 90 degrees repeats exactly between 90 and 270 degrees, and so on.

MISTAKE: Confusing the tan x graph with the sin x or cos x graphs, which are smooth, continuous waves. | CORRECTION: The tan x graph has a distinct 'S' shape between asymptotes and goes towards positive or negative infinity near these asymptotes, unlike the bounded, smooth waves of sin x and cos x.

Practice Questions

Try It Yourself

QUESTION: What is the value of tan x when x = 0 degrees? | ANSWER: 0

QUESTION: At which angle between 0 and 180 degrees is tan x undefined? | ANSWER: 90 degrees

QUESTION: Describe the general shape of the tan x graph between -90 degrees and 90 degrees. | ANSWER: It's an 'S'-shaped curve that passes through (0,0) and increases from negative infinity to positive infinity.

MCQ

Quick Quiz

Which of the following is a key characteristic of the graph of tan x?

It is a smooth, continuous wave like a sine curve.

It has vertical asymptotes where the function is undefined.

Its value always stays between -1 and 1.

It only exists for positive angles.

The Correct Answer Is:

The graph of tan x has vertical asymptotes at angles like 90 degrees and 270 degrees because the function is undefined there. Options A, C, and D are incorrect characteristics of the tan x graph.

Real World Connection

In the Real World

In cricket analytics, graphs similar to tan x (though more complex) can be used to model how a bowler's 'swing' or 'spin' changes as the ball travels, helping coaches understand player performance. Or, imagine a satellite dish needing to be pointed at a specific angle to receive signals; the math behind calculating that precise angle often involves trigonometric functions and their graphical behavior.

Key Vocabulary

Key Terms

ASYNTHOTE: A line that a curve approaches but never touches | PERIODIC FUNCTION: A function whose values repeat at regular intervals | UNDEFINED: A value that cannot be calculated, often leading to a break in the graph | TRIGONOMETRY: The study of relationships between angles and sides of triangles

What's Next

What to Learn Next

Next, you should explore the graphs of sin x and cos x. Understanding these will help you compare and contrast the different behaviors of trigonometric functions and build a complete picture of how angles relate to their values.