## 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.

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

0 | 0 | 0 |

π/6 | 0.50 | -2.00 |

π/4 | 0.71 | -2.84 |

π/3 | 0.87 | -3.48 |

π/2 | 1.00 | -4.00 |

2π/3 | 0.87 | -3.48 |

3π/4 | 0.71 | -2.84 |

5π/6 | 0.50 | -2.00 |

π | 0 | 0 |

7π/6 | -0.50 | 2.00 |

5π/4 | -0.71 | 2.84 |

4π/3 | -0.87 | 3.48 |

3π/2 | -1.00 | 4.00 |

5π/3 | -0.87 | 3.48 |

7π/4 | -0.71 | 2.84 |

11π/6 | -0.50 | 2.00 |

2π | 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 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.

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

0 | 1.00 | 3.00 |

π/6 | 0.87 | 2.61 |

π/4 | 0.71 | 2.13 |

π/3 | 0.50 | 1.50 |

π/2 | 0 | 0 |

2π/3 | -0.50 | -1.50 |

3π/4 | -0.71 | -2.13 |

5π/6 | -0.87 | -2.61 |

π | -1.00 | -3.00 |

7π/6 | -0.87 | -2.61 |

5π/4 | -0.71 | -2.13 |

4π/3 | -0.50 | -1.50 |

3π/2 | 0 | 0 |

5π/3 | 0.50 | 1.50 |

7π/4 | 0.71 | 2.13 |

11π/6 | 0.87 | 2.61 |

2π | 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 ± 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 4x | y = 2 sin 4x |
---|---|---|---|

0 | 0 | 0 | 0 |

π/12 | π/3 | 0.87 | 1.74 |

π/8 | π/2 | 1.00 | 2.00 |

π/6 | 2π/3 | 0.87 | 1.74 |

π/4 | π | 0 | 0 |

π/3 | 4π/3 | -0.87 | -1.74 |

3π/8 | 3π/2 | -1.00 | -2.00 |

5π/12 | 5π/3 | -0.87 | -1.74 |

π/2 | 2π | 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 ± 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 2x | y = 3 cos 2x |
---|---|---|---|

0 | 0 | 1.00 | 3.00 |

π/12 | π/6 | 0.87 | 2.61 |

π/8 | π/4 | 0.71 | 2.13 |

π/6 | π/3 | 0.50 | 1.50 |

π/4 | π/2 | 0 | 0 |

π/3 | 2π/3 | -0.50 | -1.50 |

3π/8 | 3π/4 | -0.71 | -2.13 |

5π/12 | 5π/6 | -0.87 | -2.61 |

π/2 | π | -1.00 | -3.00 |

7π/12 | 7π/6 | -0.87 | -2.61 |

5π/8 | 5π/4 | -0.71 | -2.13 |

2π/3 | 4π/3 | -0.50 | -1.50 |

3π/4 | 3π/2 | 0 | 0 |

5π/6 | 5π/3 | 0.50 | 1.50 |

7π/8 | 7π/4 | 0.71 | 2.13 |

11π/12 | 11π/6 | 0.87 | 2.61 |

π | 2π | 1.00 | 3.00 |

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