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 2Ï€. 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 2Ï€ are shown in the table below.

values of sin x corresponding to different values of x from 0 to 2Ï€

Clearly, sin x is a periodic function.

The following points regarding the changes in sin x as x increases from 0 to 2Ï€ 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 3Ï€/2 to 2Ï€, the value of sin x continuously increases from -1 to 0.

The graph of y = sin x is shown below.

Graph of y = sin x

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 2Ï€ are shown in the table below.

values of y = cos x corresponding to different values of x from 0 to 2Ï€

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 3Ï€/2 to 2Ï€, the value of cos x also increases continuously from 0 to 1.

The graph of y = cos x is shown below.

Graph of y = cos x

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 2Ï€ are shown in the table below.

values of tan x for different values of x lying between 0 and 2Ï€

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 Ï€/2 < x ≤ Ï€, 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 3Ï€/2 < x ≤ 2Ï€, the value of tan x continuously increases from – ∞ to 0.

The graph of tan x is shown here.

Graph of y = tan x

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 2Ï€ as -4 sin (2Ï€ + 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.

x (in radians)sin xy = -4 sin x
000
Ï€/60.50-2.00
Ï€/40.71-2.84
Ï€/30.87-3.48
Ï€/21.00-4.00
2Ï€/30.87-3.48
3Ï€/40.71-2.84
5Ï€/60.50-2.00
Ï€00
7Ï€/6-0.502.00
5Ï€/4-0.712.84
4Ï€/3-0.873.48
3Ï€/2-1.004.00
5Ï€/3-0.873.48
7Ï€/4-0.712.84
11Ï€/6-0.502.00
2Ï€00

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

graph of y = -4 sin x

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 2Ï€.

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.

x (in radians)cos xy = 3 cos x
01.003.00
Ï€/60.872.61
Ï€/40.712.13
Ï€/30.501.50
Ï€/200
2Ï€/3-0.50-1.50
3Ï€/4-0.71-2.13
5Ï€/6-0.87-2.61
Ï€-1.00-3.00
7Ï€/6-0.87-2.61
5Ï€/4-0.71-2.13
4Ï€/3-0.50-1.50
3Ï€/200
5Ï€/30.501.50
7Ï€/40.712.13
11Ï€/60.872.61
2Ï€1.003.00

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

graph of y = 3 cos x

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-

x (in radians)4x (in radians)sin 4xy = 2 sin 4x
0000
Ï€/12Ï€/30.871.74
Ï€/8Ï€/21.002.00
Ï€/62Ï€/30.871.74
Ï€/4Ï€00
Ï€/34Ï€/3-0.87-1.74
3Ï€/83Ï€/2-1.00-2.00
5Ï€/125Ï€/3-0.87-1.74
Ï€/22Ï€00

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

graph of y = 2 sin 4x

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.

x (in radians)2x (in radians)cos 2xy = 3 cos 2x
001.003.00
Ï€/12Ï€/60.872.61
Ï€/8Ï€/40.712.13
Ï€/6Ï€/30.501.50
Ï€/4Ï€/200
Ï€/32Ï€/3-0.50-1.50
3Ï€/83Ï€/4-0.71-2.13
5Ï€/125Ï€/6-0.87-2.61
Ï€/2Ï€-1.00-3.00
7Ï€/127Ï€/6-0.87-2.61
5Ï€/85Ï€/4-0.71-2.13
2Ï€/34Ï€/3-0.50-1.50
3Ï€/43Ï€/200
5Ï€/65Ï€/30.501.50
7Ï€/87Ï€/40.712.13
11Ï€/1211Ï€/60.872.61
Ï€2Ï€1.003.00

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

graph of y = 3 cos 2x

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