What is a Graph of y = sin(x)?

S3-SA5-0327

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The graph of y = sin(x) is a wave-like curve that shows how the sine value changes as the angle 'x' changes. It's a visual representation of the sine function, starting from 0, going up to 1, down to -1, and then back to 0, repeating this pattern.

Simple Example

Quick Example

Imagine a swing going back and forth in a park. If you plot its height above the ground over time, it would look like a smooth, repeating wave. The graph of y = sin(x) behaves similarly, showing a smooth, repeating up-and-down motion.

Worked Example

Step-by-Step

Let's plot some points for y = sin(x) to see its shape.

Step 1: Choose some common angle values for 'x' in degrees: 0, 30, 90, 150, 180, 210, 270, 330, 360.
---Step 2: Find the sine value (y) for each 'x'. Remember sin(0) = 0, sin(90) = 1, sin(180) = 0, sin(270) = -1, sin(360) = 0.
---Step 3: For x = 0 degrees, y = sin(0) = 0. So, point (0, 0).
---Step 4: For x = 90 degrees, y = sin(90) = 1. So, point (90, 1).
---Step 5: For x = 180 degrees, y = sin(180) = 0. So, point (180, 0).
---Step 6: For x = 270 degrees, y = sin(270) = -1. So, point (270, -1).
---Step 7: For x = 360 degrees, y = sin(360) = 0. So, point (360, 0).
---Step 8: If you plot these points (and others like (30, 0.5), (150, 0.5), etc.) on a graph and connect them with a smooth curve, you will get the wave-like shape of y = sin(x).

Why It Matters

Understanding the graph of y = sin(x) is crucial for fields like Physics, where it describes waves (sound, light, water) and oscillations. Engineers use it to design electrical circuits and predict how signals will behave. In Data Science, understanding periodic patterns helps analyze seasonal trends in data, like daily temperature changes or monthly sales.

Common Mistakes

MISTAKE: Thinking the graph is a straight line or a V-shape. | CORRECTION: The graph of y = sin(x) is always a smooth, continuous wave, not straight lines or sharp corners.

MISTAKE: Forgetting that sine values can be negative. | CORRECTION: The sine function goes from +1 to -1. The graph goes above the x-axis (positive y) and below the x-axis (negative y).

MISTAKE: Not understanding that the pattern repeats. | CORRECTION: The graph of y = sin(x) is periodic, meaning its wave shape repeats every 360 degrees (or 2*pi radians).

Practice Questions

Try It Yourself

QUESTION: What is the maximum value of y on the graph of y = sin(x)? | ANSWER: 1

QUESTION: At what angle (between 0 and 360 degrees) does the graph of y = sin(x) first cross the x-axis after starting from x=0? | ANSWER: 180 degrees

QUESTION: If the graph of y = sin(x) is plotted from x = -90 degrees to x = 450 degrees, how many times will it touch the x-axis? | ANSWER: 4 times (at 0, 180, 360, and 450 degrees, as -90 is not touching the axis)

MCQ

Quick Quiz

Which of these describes the graph of y = sin(x)?

A straight line passing through the origin

A V-shaped curve

A repeating wave that goes between -1 and 1

A parabola opening upwards

The Correct Answer Is:

The graph of y = sin(x) is a characteristic wave pattern that oscillates between its maximum value of 1 and minimum value of -1. It is not a straight line, V-shape, or parabola.

Real World Connection

In the Real World

The graph of y = sin(x) helps us understand how electricity flows in our homes. The alternating current (AC) supplied by power grids, like those from NTPC, follows a sine wave pattern. This ensures that the voltage goes up and down smoothly, allowing our fans, lights, and phones to work efficiently without sudden shocks.

Key Vocabulary

Key Terms

SINE FUNCTION: A trigonometric function that relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse. | PERIODIC: A function or graph that repeats its values in regular intervals. | AMPLITUDE: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position (for sin(x) it is 1). | OSCILLATION: The repetitive variation, typically in time, of some measure about a central value.

What's Next

What to Learn Next

Now that you understand the graph of y = sin(x), you're ready to explore the graphs of y = cos(x) and y = tan(x)! These are also wave-like but start at different points and have unique properties, which are important for advanced trigonometry and physics.