Table of Contents

## 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.

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 Ï€/2, the value of sin x gradually increases from 0 to 1.

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

(c) In the region Ï€ to 3Ï€/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.

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 Ï€/2. At even multiples of Ï€/2, the value of sin x is zero. As sin (2Ï€ + x) = sin x, the period of sin x is 2Ï€ and the portion of the graph from 0 to 2Ï€ 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.

From the table, the following points can be noted.

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

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

(c) As x increases continuously from Ï€ to 3Ï€/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.

The graph of y = cos x for 0 â‰¤ x â‰¤ 2Ï€ will be repeated in the intervals 2Ï€ â‰¤ x â‰¤ 4Ï€, 4Ï€ â‰¤ x â‰¤ 6Ï€ and son on.

It will also be repeated in -2Ï€ â‰¤ x â‰¤ 0, -4Ï€ â‰¤ x â‰¤ -2Ï€ 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.

An investigation into the table reveals the following facts.

(a) As x increases continuously in 0 â‰¤ x â‰¤ Ï€/2, the value of tan x increases continuously from 0 to âˆž.

(b) With the increase of x in Ï€/2 < x â‰¤ Ï€, the value of tax x continuously increases from – âˆž to 0.

(c) As x increases continuously in Ï€ â‰¤ x < 3Ï€/2, the value of tan x continuously increases from 0 to âˆž.

(d) As x increases continuously in 3Ï€/2 < x â‰¤ 2Ï€, the value of tan x continuously increases from – âˆž to 0.

The graph of tan x is shown here.

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

### 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 â‰¤ x â‰¤ 2Ï€ is given below.

x (in radians) | sin x | y = -4 sin x |
---|---|---|

0 | 0 | 0 |

Ï€/6 | 0.50 | -2.00 |

Ï€/4 | 0.71 | -2.84 |

Ï€/3 | 0.87 | -3.48 |

Ï€/2 | 1.00 | -4.00 |

2Ï€/3 | 0.87 | -3.48 |

3Ï€/4 | 0.71 | -2.84 |

5Ï€/6 | 0.50 | -2.00 |

Ï€ | 0 | 0 |

7Ï€/6 | -0.50 | 2.00 |

5Ï€/4 | -0.71 | 2.84 |

4Ï€/3 | -0.87 | 3.48 |

3Ï€/2 | -1.00 | 4.00 |

5Ï€/3 | -0.87 | 3.48 |

7Ï€/4 | -0.71 | 2.84 |

11Ï€/6 | -0.50 | 2.00 |

2Ï€ | 0 | 0 |

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 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 â‰¤ x â‰¤ 2Ï€ is given below.

x (in radians) | cos x | y = 3 cos x |
---|---|---|

0 | 1.00 | 3.00 |

Ï€/6 | 0.87 | 2.61 |

Ï€/4 | 0.71 | 2.13 |

Ï€/3 | 0.50 | 1.50 |

Ï€/2 | 0 | 0 |

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€/2 | 0 | 0 |

5Ï€/3 | 0.50 | 1.50 |

7Ï€/4 | 0.71 | 2.13 |

11Ï€/6 | 0.87 | 2.61 |

2Ï€ | 1.00 | 3.00 |

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 Â± 2nÏ€) = 2 sin 4 (x Â± nÏ€/2). Thus, y = 2 sin 4x is periodic with period Ï€/2. It is, thus, sufficient to sketch the graph only for the interval 0 â‰¤ x â‰¤ Ï€/2. In other intervals of length Ï€/2, the nature of the graph would be exactly similar to that in interval 0 â‰¤ x â‰¤ Ï€/2.

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

x (in radians) | 4x (in radians) | sin 4x | y = 2 sin 4x |
---|---|---|---|

0 | 0 | 0 | 0 |

Ï€/12 | Ï€/3 | 0.87 | 1.74 |

Ï€/8 | Ï€/2 | 1.00 | 2.00 |

Ï€/6 | 2Ï€/3 | 0.87 | 1.74 |

Ï€/4 | Ï€ | 0 | 0 |

Ï€/3 | 4Ï€/3 | -0.87 | -1.74 |

3Ï€/8 | 3Ï€/2 | -1.00 | -2.00 |

5Ï€/12 | 5Ï€/3 | -0.87 | -1.74 |

Ï€/2 | 2Ï€ | 0 | 0 |

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 Â± 2nÏ€) = 3 cos 2(x Â± nÏ€). Thus, the function 3 cos 2x is periodic with period n = Ï€. It is, thus sufficient to sketch the graph of the function y = 3 cos 2x only for the interval 0 â‰¤ x â‰¤ Ï€. For other intervals of length Ï€, the graph would be exactly the same as for the interval 0 â‰¤ x â‰¤ Ï€. To sketch the graph, the table for y = 3 cos 2x is constructed as follows.

x (in radians) | 2x (in radians) | cos 2x | y = 3 cos 2x |
---|---|---|---|

0 | 0 | 1.00 | 3.00 |

Ï€/12 | Ï€/6 | 0.87 | 2.61 |

Ï€/8 | Ï€/4 | 0.71 | 2.13 |

Ï€/6 | Ï€/3 | 0.50 | 1.50 |

Ï€/4 | Ï€/2 | 0 | 0 |

Ï€/3 | 2Ï€/3 | -0.50 | -1.50 |

3Ï€/8 | 3Ï€/4 | -0.71 | -2.13 |

5Ï€/12 | 5Ï€/6 | -0.87 | -2.61 |

Ï€/2 | Ï€ | -1.00 | -3.00 |

7Ï€/12 | 7Ï€/6 | -0.87 | -2.61 |

5Ï€/8 | 5Ï€/4 | -0.71 | -2.13 |

2Ï€/3 | 4Ï€/3 | -0.50 | -1.50 |

3Ï€/4 | 3Ï€/2 | 0 | 0 |

5Ï€/6 | 5Ï€/3 | 0.50 | 1.50 |

7Ï€/8 | 7Ï€/4 | 0.71 | 2.13 |

11Ï€/12 | 11Ï€/6 | 0.87 | 2.61 |

Ï€ | 2Ï€ | 1.00 | 3.00 |

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

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