Math DoubtsTrigonometryTrigonometric functions Graphs

# Sine Graph

How to construct Sine Graph and Analysis of functionality of sine function from sine graph

A graph between angles and the values of sine function at same angles is defined Sine Graph.

According to Trigonometry, sine function is originally defined from a right angled triangle as a ratio of opposite side to hypotenuse at the respective angle. So, the ratio between length of the opposite side and length of the hypotenuse is the sine value at the corresponding angle of the right angled triangle.

Using this principle, values of sine function can be evaluated for all the angles. Plotting a graph between angles and sine values at the respective angles gives us a smooth curve known a sine graph. It is most helpful in trigonometry to understand the functionality of this trigonometric ratio.

In order to plot a sine graph, take angles in horizontal $x$-axis and corresponding sine values in vertical $y$-axis. Sine graph follows a pattern which can be analysed in four different stages.

## Analysis

Evaluate the sine function's values from 0° to 90° and also evaluate sine values from 90° to 180°. Construct a graph by using the angles and corresponding sine values.

The sine function gives zero at 0° and the values of this function are increased from 0 to 1 as the value of angle is increased. It follows the same rule until 90 degrees where it gives its maximum value 1. For this reason, the graph displays raising during 0° to 90° region.

Thereafter, the functionality of this function becomes opposite from 90 degrees onwards. It gives positive values but decrement in values can be observed during the interval of 90° to 180°. In other words, the values of sine function are decreased from 1 to 0. For this reason, the graph displays falling during 90° to 180°. It follows same rule until 180° where it gives a value same as 0°.

Determine the values of sine function from 180° to 270° and also determine sine function's values from 270° to 360°. Now, plot a graph by using these values.

The function starts giving negative values from 180 degrees onwards and continuous the same till 360°. However, there is a slight variation from 180° to 270° region and also 270° to 360° region.

Sine function gives decreased values from 0 to -1 during 180 to 270 degrees region. In other words, it gives 0 at 180° and gives decreased values till 270°, where it gives its minimum value -1. For this reason, decrement in sine graph can be observed from 180° to 270°.

However, it changes its functionality from 270° onwards. It shows increment in its values and continues the same till 360°, where it gives 0. The increment in sine values can be observed in the graph during the region of 270° to 360°.

The functionality of the sine function from 0° to 360° is repeated for every next 360° interval because whatever values the sine function has given from 0° to 360°, the same values are given by this function for every 360° interval.

Now, it is time to analyse the functionality of the sine function for the negative angles. It understands you how sine function responds for negative angles.

Calculate the values of sine function from 0° to -90° and also calculate the values of sine function from -90° to -180°. Now plot a graph by using the negative angles and corresponding sine function's values.

It gives negative values for negative angles from 0 to -180 degrees because sine is originally an odd function. It gives values from 0 to -1 as the value of angle is decreased from 0 to -90 degrees. At -90°, it gives its minimum value -1. Sine graph clearly displays the decrement in its values from 0° to -90°.

However, it changes its functionality from -90 degrees onwards. Even though the angle is decreased from -90° to -180°, the value of sine function is increased from -1 to 0. The same functionality is continued in sine function till the value of -180°. The sine graph also displayed increment in its values during -90° to -180°.

Determine the values of sine function from -180° to -270° and also determine the values of sine function from -270° to -360°. Also, construct a graph by taking angles and corresponding sine values.

Sine function starts giving positive values for negative angles from -180° onwards and it is continued till -360°. As the angle is decreased from -180° to -270°, the function returns the values from 0 to 1. The increment in sine function's values can be observed in the sine graph from -180° to -270°.

But, sine function changes its functionality and starts giving decreased values from -270° onwards and continues the same functionality till -360°. In other words, it gives decreased values from 1 to 0 during -270° to -360°. It can be observed in the graph.

The pattern of sine function from 0 to -360° is repeated for every 360 degrees interval because whatever values the function has given from 0° to -360°, the same values are given by the function for the next 360° interval and it is continued infinitely.

As explained in above four steps, construct a sine graph by taking angles in horizontal $x$-axis and corresponding sine function's values in vertical $y$-axis. It gives us an oscillating smooth wave style curve infinitely.

Sine graph continues its oscillation between -1 to 1 values and it is continued infinitely in both positive and negative directions.