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Graphs of Circular Functions


The equation x2 + y2 = 1 forms the Circle in Rectangular Coordinate System. This graph is known as Unit Circle it has radius one with the center at its origin i.e. (0, 0). The Domain of the trigonometric function is the Set of angles and their ranges are the set of real Numbers. Circular Functions are defined in a way such that Sets of angles are domain which correspond to the measures of the angles of the analogous trigonometric function, like in analog Trigonometry the range of these Trigonometric Functions are also the set of Real Numbers and these type of functions are termed as Circular Functions, here the Radian Measure of angle is determined by the length of arc of circle. In other words when Trigonometric Functions are defined using unit circle then they directly head towards these graphs of the circular functions.
Lets plot the graphs of circular functions with unit circle equation x2 + y2 = 1 as we can see in the figure:

Point A (1, 0) is located at the Intersection of x - axis and of the unit circle. Let ‘q’ be the real number. Graphing circular functions is done by Initiating from Point ‘A’ and measuring │q│ units along the unit of the circle in anti clockwise direction. Here q > 0 and in clockwise direction if q < 0 and it ends up at point p(x, y). So other remaining circular function such as Tangent, cotangent, secant, cosecant can be defined in the terms of sine and cosine.
Sin q = y,
cos q = x,
Therefore rest of the functions will be:
tan q = sin q/ cos q ; cos q ≠ 0,
cot q = cos q / sin q ; sin q ≠ 0,
sec q = 1/ cos q ; cos q ≠ 0,
cosec q = 1/ sin q ; sin q ≠ 0.

Characteristic of Cosine Graph

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There are different types of the graphs which we study under the subject of the Math especially in the topic of the Trigonometry for example the graph of the sine function, the graph of the cosine function, the graph of the Tangent function and also the graphs of the Logarithm function and the Exponential Function and many other graphs. It is very important to know the characteristic of any graph which we study. So in this article we will discuss some of the important properties of cosine graph.

The equation of the graph of the cosine function is y = cos x. So let us start discussing about the properties of cosine graph with Domain and The Range of the graph. The domain of the cosine graph includes all the numbers which exist on the line of the Real Numbers whereas the range of the cosine graph consists of all the real numbers existing between -1 and 1 including the numbers -1 and 1 also.

The other properties of cosine graph are the x intercepts and the y intercepts. The graph of the cosine function intersects the horizontal axis at -2700, at -900, at 900, at 2700 and at the 4500. These values in the degree have a difference of 1800 in them while each of these values can be represented in the form of 900 + any multiple of the 1800 where the 900 is the first intercept on the x axis which is positive. The graph of the cosine function intersects the vertical axis at y = 1 that is when the value of the x is zero because y = cos (0) = 1.
There are also 2 more characteristics of cosine graph that is the amplitude which is equal to 1 and the Period which is equal to 3600.

Characteristic of Sine Graph

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It is very common for us to study about the different Types of Graph in Math. Especially very frequently we come across various types of graphs of different kinds of trigonometric Functions. For example, the graphs of the functions like sine function, tangent function and cosine function are the ones on which we usually pay much of the attention because the graph of many other functions depend on the graph of these important functions. Let’s see some important properties of sine graph.
First of all we should know that, equation of graph of the sine function is being written as y = sin x. Now let us start discussing the properties of sine graph. Graph of sine function passes through the Point of origin represented by ( 0, 0 ). The graph of this function is continuous along the horizontal axis that is the x axis and it attains a maximum value which is equal to 1 while a minimum value which is equal to -1 on the vertical axis that is y axis.
We can see from the graph of the sine function that the Domain of this function consists of all the Numbers existing on the real line whereas its range is from -1 to 1 including -1 and 1 also. It can be observed from graph of sine function that sine function is a kind of a function which is periodic due to the reason that it tends to repeat itself over the intervals which are generally known as periods of graph.

Steps to Graph a Basic Sine or Cosine Function

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First of all lets go through sine and cosine function and then we will see steps to graph a basic sine or cosine function. A sine function is basically a function which is a function of angle. It is related to the sides of triangle and angles of a Right Triangle. Let us take a a Right Angle triangle with base AB , Height BC and Hypotenuse AC.

Now the sin Θ = opposite sides of triangle / Hypotenuse,
Sin Θ is similar to Θ.
Θ = BC / AC.
Similarly cosine function is equal to adjusant side / Hypotenuse.
Here cos Θ = AB / AC.
Cosine Function: Cosine function is known as complementary angle of sine.

Steps to draw sine function graph are:
1. First draw two axes i.e. x and y- axis on a plane and mark two points on y- axis as 2 on positive y- axis and -2 on negative y axis . On x- axis draw the degrees as 0 , 90 , 180, 270 and 360. Similarly draw on left side of Negative x- axis.
2. Now since we know that sin 0 is 0 (zero) therefore we will plot sine curve from zero. Now sin 900 is 1 therefore we will mark it as on positive y- axis i.e. 2.
3. Like this we will continue and plot oscillations for the given degree scale.
The graph of Cosine function:

1. In this graph we will plot same 'x' and y- axis with degrees marked with a scale of 90 degree.
2. After that we will plot the curve since, cos 0 is 1 therefore we will start the curve from a positive value now, next cos 90 is zero therefore we will move curve to zero.
3. Similarly we will plot the graph after looking for all angles on x- axis.

Graphs of Sine and Cosine Functions

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Graphs of sine and cosine Functions can be used in plotting daily temperature in a year, in Graphing the cyclic commodity prices, in finding the heart beat rates on the ECG machines. The graph of Sine and cosine functions can be plotted easily. Sine function is a function used to model any periodic phenomenon which can be of sound waves, intensity of sunlight etc. We can also plot sound waves with help of graph of sine and cosine functions.

A simple graph of sine function is plotted above, with help of this graph we can understand basic concept of the sine function. On X-axis of graph we have radians marked as 1/2π , π, 3/2 π and 2π. Here value of 'π' is 180 degree. Therefore values of above radians are 900, 1800, 2700 and 3600. Similarly on Y- axis we have the value one and minus one. Here in this curve of sine starts from the origin 00, the most important thing is that frequency of oscillation remains the same throughout the oscillation.

Cosine Graph: A cosine function is defined as trigonometric function of angles. It is used in modeling periodic formulas; it is also used in computing various lengths & angles.

The graph of cosine function does not start from origin but from value 1 because cosine 00 that is 1 and cosine 900 that is zero. This is the main difference between sine and cosine graphs. x- axis is marked as with the degree values, it can also be marked as radian values of 'π' like 1/2π, π, 3/2 π etc. y- axis is marked as values of cosine 1 and -1.
This is the process of graphing sine and cosine function.

Graphs of Basic Circular Functions

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A circular or periodic function F (X) can be defined as a function in which for every real number 'X' in the Domain of F (X), every Integer 'N', and some positive real number 'P'. Smallest possible value of 'P' represents the Period of function. Graphs of basic circular Functions can be drawn as follows:
Graph of sine function can be drawn as follows:

Characteristics of the graph:
Graph is continuous over complete domain, (- infinity, infinity).
In graph Intersection points with x - axis are of the form n∏ where, 'n' is real.
Period of sine function is 2 ∏.
About the origin we get Symmetry in graph. So, sine function can be said as odd function.
For all values of 'x' in the domain, we have sin (-x) = - sin x.
Graph of cosine function can be drawn as follows:
Characteristics of graph:

This graph is also continuous over complete domain, (- infinity, infinity).
In graph intersection points with x - axis are of form (2n + 1) ∏ / 2. Where, 'n' is real.
Period of cosine function is also 2 ∏.
About y - axis we get symmetry in graph. So, cosine function can be said as even function.
For all 'x' in domain, cos (-x) = cos x.

Value of A goes from

Sin A

Cos A


0 to pi / 2

Increases from 0 to 1

Decreases from 1 to 0


Pi / 2 to pi

Decreases from 1 to 0

Decreases from 0 to -1


Pi to 3pi / 2

Decreases from 0 to -1

Increases from -1 to 0


3pi / 2 to 2pi

Increases from -1 to 0

Increases from 0 to 1

Graphs of inverse Circular Functions

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Six Trigonometric Functions are: sine, cosine, tangent, cotangent, secant and cosecant. Their corresponding inverse functions can be written as arc sine, arc cosine, arc Tangent, arc cotangent, arc secant and arc cosecant. These functions are also periodic in nature. Graphs of inverse Circular Functions can be drawn as follows:

1. Graph of arc sin A

Yields positive values in I quadrant and negative values in IV quadrant.
2. Graph of arc cos A

In I quadrant positive values are found and in II quadrant negative values are found.

3. Graph of arc tan A

A = tan-1 y
The graph cuts the origin, the common Intersection Point for both graph and x and y axes. It lies in I and III quadrants.

4. Graph of arc cosec A

A = cosec-1 y

The graph extends to negative and positive infinity along x axis and lies in the I and the III quadrants only.

5. Graph of arc sec A

A = sec-1 y

The graph lies in the I and II quadrants only.

6. Graph of arc cot A

A = cot-1 y

The graph extends to negative and positive infinity along x axis and lies in the I and the II quadrants only.
The table below shows the Domain and ranges of the six Inverse Trigonometric Functions:

Function name




Inverse cosecant


(- ∞, ∞)

[- pi / 2, 0) or (0, pi / 2]

Inverse cosine


[-1, 1]

[0, pi]

Inverse cotangent


(- ∞, ∞)

(- pi / 2, 0) or (0, pi / 2]

Inverse secant


(- ∞, ∞)

[0, pi / 2) or (pi / 2, pi]

Inverse sine


[-1, 1]

[-pi / 2, pi / 2]

Inverse tangent


(- ∞, ∞)

(-pi / 2, pi / 2)