Sin x Graph Explained with Key Properties

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.

1. Draw a Y-axis with numbering 0,1,2.. etc.

2. Draw an X-axis with notations π/2, π, 3π/2, 2π.. Etc.

3. The sinx curve intersects the X-axis at 0, π, 2π, 3π… and so on.

4. The sinx curve intersects the Y-axis only at the origin. 

5. 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

6. 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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