Graphs of Simple Trigonometric Functions

Graphs of Simple Trigonometric Functions:

We know that trigonometric functions are periodic. The values of a trigonometric function (ratio) are repeated after each interval of 2π. This property of periodicity of trigonometric functions helps one to draw the graphs of trigonometric functions over large intervals.

Graph of y = sin x:

The values of sin x corresponding to different values of x from 0 to 2π are shown in the table below.

Clearly, sin x is a periodic function.

The following points regarding the changes in sin x as x increases from 0 to 2π radians are to be noted carefully.

(a) A x increases from 0 to π/2, the value of sin x gradually increases from 0 to 1.

(b) In the region π/2 to π, the value of sin x gradually decreases from 1 to 0.

(c) In the region π to 3π/2, the value of sin x gradually decreases from 0 to -1.

(d) As x increases continuously from 3π/2 to 2π, the value of sin x continuously increases from -1 to 0.

The graph of y = sin x is shown below.

It is clear from the graph that it is a continuous graph extending on either side in the form of a symmetrical wave. The maximum value is +1 and the minimum value is -1. The maximum and the minimum values occur at odd multiples of π/2. At even multiples of π/2, the value of sin x is zero. As sin (2π + x) = sin x, the period of sin x is 2π and the portion of the graph from 0 to 2π goes on being repeated on either side.

Graph of y = cos x:

The values of y = cos x corresponding to different values of x from 0 to 2π are shown in the table below.

From the table, the following points can be noted.

(a) As x increases continuously from 0 to π/2, the value of cos x diminishes continuously from 1 to 0.

(b) In the region π/2 to π, the value of cos x diminishes gradually from 0 to -1.

(c) As x increases continuously from π to 3π/2, the value of cos x also increases continuously from -1 to 0.

(d) As x increases further from 3π/2 to 2π, the value of cos x also increases continuously from 0 to 1.

The graph of y = cos x is shown below.

The graph of y = cos x for 0 ≤ x ≤ 2π will be repeated in the intervals 2π ≤ x ≤ 4π, 4π ≤ x ≤ 6π and son on.

It will also be repeated in -2π ≤ x ≤ 0, -4π ≤ x ≤ -2π and so on.

Graph of y = tan x:

The values of tan x for different values of x lying between 0 and 2π are shown in the table below.

An investigation into the table reveals the following facts.

(a) As x increases continuously in 0 ≤ x ≤ π/2, the value of tan x increases continuously from 0 to ∞.

(b) With the increase of x in π/2 < x ≤ π, the value of tax x continuously increases from – ∞ to 0.

(c) As x increases continuously in π ≤ x < 3π/2, the value of tan x continuously increases from 0 to ∞.

(d) As x increases continuously in 3π/2 < x ≤ 2π, the value of tan x continuously increases from – ∞ to 0.

The graph of tan x is shown here.

Note: The function y = tan x is periodic with period π radians. So, it is sufficient to sketch the graph of y = tan x for the interval 0 ≤ x ≤ π. For each of the other intervals of length π, the graph would be exactly the same as that in the interval 0 ≤ x ≤ π.

Graph of y = a sin x:

Let us take a = -4. The function then becomes y = -4 sin x. It is a periodic function with period 2π as -4 sin (2π + x) = -4 sin x.

The values of -4 sin x at different values of x are just -4 times the values of sin x for the corresponding values of x. The table for y = -4 sin x for various values of x in the interval 0 ≤ x ≤ 2π is given below.

The graph of y = -4 sin x is shown below.

Graph of y = a cos x:

Let us take a = 3 here. Then, the function becomes y = 3 cos x, which is evidently a periodic function with period 2π.

The values of 3 cos x at different values of x would be just 3 times those of cos x for the corresponding values of x.

The table showing the values of 3 cos x at different values of x in the interval 0 ≤ x ≤ 2π is given below.

The graph of y = 3 cos x is given below.

Graph of y = a sin bx:

Let a = 2 and b = 4. The function then becomes y = 2 sin 4x. The function can be written as y = 2 sin 4x = 2 sin (4x ± 2nπ) = 2 sin 4 (x ± nπ/2). Thus, y = 2 sin 4x is periodic with period π/2. It is, thus, sufficient to sketch the graph only for the interval 0 ≤ x ≤ π/2. In other intervals of length π/2, the nature of the graph would be exactly similar to that in interval 0 ≤ x ≤ π/2.

To sketch the graph, the table for y = 2 sin 4x is constructed as follows-

The graph of y = 2 sin 4x is shown here.

Graph of y = a cos bx:

Let us take a = 3, b = 2. Then, the function becomes y = 3 cos 2x.

Now, 3 cos 2x = 3 cos (2x ± 2nπ) = 3 cos 2(x ± nπ). Thus, the function 3 cos 2x is periodic with period n = π. It is, thus sufficient to sketch the graph of the function y = 3 cos 2x only for the interval 0 ≤ x ≤ π. For other intervals of length π, the graph would be exactly the same as for the interval 0 ≤ x ≤ π. To sketch the graph, the table for y = 3 cos 2x is constructed as follows.

The graph of y = 3 cos 2x is shown below.