How to Sketch and Analyze the Sinx Graph Step by Step
When we write trigonometric ratios, they can also be represented in a graphical format where the functions of the variable are the measure of the angle. These angles can be either in the degree or radians.
Let’s talk about the Sinx graph. The graph of sin function is represented by the equation: y=Sin(x).
The graphical representation of this graph is:
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Properties Of The y = Sinx Graph Or The Sine Function
Now that we know how the graph of sin looks, let’s have a look at its properties. These values of the sine function will help you while solving problems relating to the Sinx graph.
Domain - It is the set of all values that can be used as input for which the function is defined. All values that can be used as input for the function are in the domain of the function. The domain of the sine function is from (-∞,∞) .
Range - It is defined as the set of all values that are outputs of the function for all the values in the domain of the function. The range of the y Sinx Graph is [-1,1]. It can also be represented as -1y1.\[-1\leq{y}\leq1\]
Y-Intercept- It represents the coordinates of the point where the sin x graph intersects with the y-axis. The y-intercept of the y sinx graph is (0,0) which means that the graph passes through the Y-axis at the origin. Therefore, when the value of x=0, the value of Y is also 0.
X-Intercept - It is defined as the coordinates of the points where the Sin x graph intercepts the X-axis. The x-intercept of the Sinx graph is nπ, where n is an integer. It means that the graph cuts the x-axis at equal intervals of π. Here, n is an integer and therefore can also take negative values.
Period: The period of the y sinx graph is 2π. This means that after an interval of 2π, the graph repeats itself.
Continuity- The graph of Sine is continuous on (-∞,∞) which states the function is continuous everywhere.
Symmetry- The y = Sinx Graph is symmetrical at the origin which means that it is an odd function. If a graph is symmetrical at the Y-axis it is termed as an even function. Even function is represented by f(x)=f(-x) and odd function is represented by f(x)=-f(x).
Drawing A Sine Function
The y Sin x graph is representative of a periodic function y = Sin(x) with a period of 2π. The graph is continuous on (-∞,∞) so we will draw the graph in the interval [0,2π]. Follow these steps to draw the graph of the sine function.
Draw a Y-axis with numbering 0,1,2.. etc.
Draw an X-axis with notations π/2, π, 3π/2, 2π.. Etc.
The sinx curve intersects the X-axis at 0, π, 2π, 3π… and so on.
The sinx curve intersects the Y-axis only at the origin.
Let’s find the values of y for values of x
At x=0, y=Sin(0)=0
At x=π/2, y=Sin(π/2)=1
At x=π, y=Sin(π)=0
Draw the sinx curve by joining the points (0,0), (π/2,1) and (2π,0). It will look something like this:
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Other curves that follow this shape are termed “sinusoidal” after the sine function. This is also referred to as the sine curve and is commonly found in radio and electronic circuits.
Graph of Mod Sinx
The graph of mod sinx is given below:
y=|Sin(x)|
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The range of this function now changes from (-1,1) to (0,1), because the output values need to be positive as per the definition of the mode function.
One another way graph of mod sinx can be represented is if the function becomes:
y=Sin|x|
Here, the domain of the function changes from (-∞,∞)to (0,∞).
Its graph can be represented as :
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Graph of xSinx
Graph of xsinx can be obtained by:
First, we will draw the graph of y=Sin x
Let’s obtain the values of y for values of x.
f(0)=0
f(π/2)=π/2
f(π)=0
f(3π/2)=-3π/2
f(2π)=0
f(5π/2)=5π/2
And so on.
The points which lie on y=x are:
(π/2,π/2), (5π/2, 5π/2), …
The points which lie on y=-x are:
(3π/2, 3π/2), (7π/2, 7π/2),...
Also,
f(x)=f(-x), therefore the function is an even function.
An even function is one where the graph is symmetrical about the Y-axis.
The graph then comes out to be this.
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FAQs on Sinx Graph Explained: Properties, Shape & Applications
1. What are the key characteristics of the sin x graph?
The graph of the function y = sin x, also known as the sine wave or sinusoid, has several key characteristics that are fundamental to understanding trigonometry. As per the CBSE/NCERT syllabus for the 2025-26 session, these are:
- Periodic Nature: The graph repeats itself at regular intervals. The period of the sin x graph is 2π radians (or 360°), which means the shape of the graph from x = 0 to x = 2π is the same for every subsequent 2π interval.
- Domain and Range: The domain (all possible x-values) is all real numbers, written as (-∞, ∞). The range (all possible y-values) is from -1 to 1, inclusive, written as [-1, 1].
- Amplitude: The amplitude is the maximum distance from the centre line (the x-axis). For y = sin x, the amplitude is 1.
- Intercepts: The graph passes through the origin (0,0). It intercepts the x-axis at integer multiples of π (e.g., 0, π, 2π, -π). It has a maximum value of 1 at π/2 and a minimum value of -1 at 3π/2 within one cycle.
2. How do you plot the graph of y = sin x for one full cycle (0 to 2π)?
To plot one full cycle of the y = sin x graph from 0 to 2π radians, you can use five key points that correspond to the quadrantal angles. These points help define the iconic wave shape:
- At x = 0, sin(0) = 0. The point is (0, 0).
- At x = π/2 (90°), sin(π/2) = 1. This is the maximum point, (π/2, 1).
- At x = π (180°), sin(π) = 0. The graph crosses the x-axis again at (π, 0).
- At x = 3π/2 (270°), sin(3π/2) = -1. This is the minimum point, (3π/2, -1).
- At x = 2π (360°), sin(2π) = 0. The cycle completes at (2π, 0).
By plotting these five points and drawing a smooth, continuous curve through them, you create one complete period of the sine wave.
3. How is the graph of y = sin x different from the graph of y = cos x?
While both the sin x and cos x graphs are sinusoidal waves with the same shape, amplitude (1), and period (2π), their key difference is a horizontal shift, also known as a phase shift. The graph of cos x is essentially the graph of sin x shifted π/2 units to the left. This means:
- Starting Point (at x=0): The sin x graph starts at the origin (0,0), on its way up. The cos x graph starts at its maximum value (0,1).
- Symmetry: The sin x graph has origin symmetry (it is an odd function). The cos x graph has y-axis symmetry (it is an even function).
In essence, cos(x) = sin(x + π/2).
4. Why is the sin x function considered an odd function, and how does its graph show this?
The function y = sin x is an odd function because it satisfies the mathematical property f(-x) = -f(x). For the sine function, this means sin(-x) = -sin(x) for any value of x. For example, sin(-π/2) = -1, which is the negative of sin(π/2) = 1.
This property is visually represented on the graph through its symmetry about the origin. If you rotate the entire sin x graph 180° around the point (0,0), the graph will land perfectly back on itself. This is a clear graphical test for an odd function.
5. What happens to the sin x graph when you plot its absolute value, y = |sin x|?
When you take the absolute value of the sine function, y = |sin x|, any part of the graph that is below the x-axis (where sin x is negative) is reflected above the x-axis. This causes two major changes to the graph:
- Range: The original range of [-1, 1] changes to [0, 1], as the function can no longer have negative values. All the troughs of the wave are flipped upwards to become crests.
- Period: The graph now repeats itself every π radians instead of 2π. The period is effectively halved to π because the negative portion of the wave from π to 2π becomes identical to the positive portion from 0 to π.
6. How does changing the function to y = 2sin(x) affect the graph of y = sin x?
Changing the function from y = sin(x) to y = 2sin(x) primarily affects the amplitude of the graph. The number '2' in front of the sin function acts as a vertical stretch factor.
- The amplitude increases from 1 to 2. This means the graph will reach a maximum height of 2 and a minimum depth of -2.
- The graph is stretched vertically by a factor of 2.
- The period (2π) and the x-intercepts (at 0, π, 2π, etc.) remain unchanged because the horizontal properties of the graph are not affected.
7. How does the graph of y = sin(2x) differ from the standard y = sin(x) graph?
The change from y = sin(x) to y = sin(2x) affects the period of the graph. The '2' inside the function causes a horizontal compression.
- The period is halved from 2π to π. This is calculated using the formula Period = 2π/|b|, where b=2. So, Period = 2π/2 = π.
- This means the graph completes two full cycles in the same interval (0 to 2π) where the standard sin(x) graph completes only one.
- The amplitude (1) and the range [-1, 1] remain unchanged, as the vertical properties of the graph are not affected.
