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

S3-SA5-0152

Grade Level:

Class 10

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 value of sin(x) changes as 'x' (an angle) changes. It's a visual representation of the sine function, which describes repeating patterns in nature.

Simple Example

Quick Example

Imagine a swing moving back and forth in a park. If you plot the height of the swing above the ground over time, it would look very similar to the graph of y = sin(x). It goes up, comes down, passes the middle, goes down further, and then comes back up, repeating this cycle.

Worked Example

Step-by-Step

Let's plot some points for y = sin(x) for angles from 0 degrees to 360 degrees.

Step 1: Understand sine values for common angles.
sin(0 degrees) = 0
sin(90 degrees) = 1
sin(180 degrees) = 0
sin(270 degrees) = -1
sin(360 degrees) = 0

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Step 2: Take more points for a smoother curve.
sin(30 degrees) = 0.5
sin(45 degrees) = 0.707 (approx)
sin(60 degrees) = 0.866 (approx)
sin(120 degrees) = 0.866 (approx)
sin(150 degrees) = 0.5

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Step 3: Similarly, for angles between 180 and 360 degrees, sine values become negative.
sin(210 degrees) = -0.5
sin(240 degrees) = -0.866 (approx)
sin(300 degrees) = -0.866 (approx)
sin(330 degrees) = -0.5

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Step 4: Now, imagine plotting these (angle, sine value) pairs on a graph. The x-axis would represent the angle (in degrees or radians) and the y-axis would represent the sine value.

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Step 5: Connect these points smoothly. You will see a curve that starts at (0,0), goes up to a peak at (90,1), comes down to (180,0), goes further down to a trough at (270,-1), and then comes back up to (360,0).

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Step 6: This wave-like pattern is the graph of y = sin(x). It repeats every 360 degrees (or 2*pi radians).

Why It Matters

Understanding the sine wave is super important because it describes almost all kinds of waves around us! From sound waves in your music player to radio waves carrying mobile signals, and even light waves, they all follow this pattern. Engineers use it to design everything from bridges to circuits, and scientists use it to understand earthquakes and ocean tides.

Common Mistakes

MISTAKE: Confusing the graph of sin(x) with cos(x) or tan(x). | CORRECTION: Remember that sin(x) starts at 0 when x=0, goes up to 1, then down to -1, and back to 0. Cos(x) starts at 1 when x=0.

MISTAKE: Forgetting that the sine wave repeats. | CORRECTION: The sine wave is periodic; it completes one full cycle every 360 degrees (or 2*pi radians) and then the pattern repeats endlessly.

MISTAKE: Plotting angles in degrees but thinking in radians, or vice-versa, on the x-axis. | CORRECTION: Always be clear if your x-axis represents angles in degrees (0, 90, 180...) or radians (0, pi/2, pi...).

Practice Questions

Try It Yourself

QUESTION: What is the maximum value that y = sin(x) can reach? | ANSWER: 1

QUESTION: At what angle (between 0 and 360 degrees) does the graph of y = sin(x) first reach its minimum value? | ANSWER: 270 degrees

QUESTION: If the graph of y = sin(x) passes through (pi, 0), what other angle (in radians, between pi and 2*pi) will it pass through (x, 0)? | ANSWER: 2*pi

MCQ

Quick Quiz

Which of the following statements is true about the graph of y = sin(x)?

It is a straight line.

It always stays above the x-axis.

It is a wave-like curve that repeats its pattern.

It has a maximum value of 0.

The Correct Answer Is:

The graph of y = sin(x) is a periodic, wave-like curve, not a straight line. It goes both above and below the x-axis, and its maximum value is 1, not 0.

Real World Connection

In the Real World

When you hear music on your phone or radio, the sound waves travel in patterns very similar to a sine wave. Audio engineers use their knowledge of sine waves to mix music and create special sound effects. Even the electricity coming to your home from the power plant is an 'alternating current' (AC) that follows a sine wave pattern.

Key Vocabulary

Key Terms

SINE FUNCTION: A mathematical function that describes periodic oscillations or waves. | PERIODIC FUNCTION: A function whose values repeat after a fixed interval. | 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's 1. | CYCLE: One complete repetition of a periodic pattern. | RADIAN: A unit of angle, equal to the angle subtended at the center of a circle by an arc equal in length to the radius.

What's Next

What to Learn Next

Great job understanding the sine wave! Next, you should explore the graph of y = cos(x) and y = tan(x). You'll see how they are related to the sine wave and how their shapes are slightly different, which is key to understanding more complex wave phenomena.