We are aware of the trigonometric functions which are the ratios of the pair of sides of the right angle containing the acute angle of the trigonometric functions. In general any function is of the form y = f ( x ) where x is the independent variable and y the dependent variable. In a similar way we can express the trigonometric functions of the form y = sin x, y = cos x and so one, where x is an independent variable and y the dependent variable. The domain of x consists of all real values when we consider the angles in radian form. The range of the function belongs to real numbers since the trigonometric function of any angle is a real value which is the ratio of the corresponding sides of the triangle which is a real number. In this section let us see the graphs of the trigonometric functions and their properties.


Graphing the Trig function y = sin x


Let us assume the values for x and find the corresponding values for y.
The following table shows the values of y for the corresponding values of y.
x
- 2 $\pi$
-3 $\frac{\pi}{2}$
- $\pi$
- $\frac{\pi}{2}$ 0
$\frac{\pi}{2}$ $\pi$ 3 $\frac{\pi}{2}$ 2 $\pi$
y 0
1
0
-1
0
1
0
-1
0

Let us plot these points in the graph and join the points.
sinx
From the above graph we observe that
1. The curve is continuous from - 2 $\pi$ to + 2$\pi$
2. The curve passes through the origin.
3. The domain consists of all real numbers.
In the above graph Domain = { - 2 $\pi$, 2 $\pi$}
4. The graph has its maximum value y = 1 and the min value y = -1.
Therefore, the range of the function is { x : x $\epsilon$ [-1, 1 ], x is a Real number }
5. The curve reaches maximum at x = - 3 $\frac{\pi}{2}$ and $\frac{\pi}{2}$,

therefore the period of the function is = $\frac{\pi}{2}$ - (- 3 $\frac{\pi}{2}$)

= $\frac{\pi}{2}$ + 3 $\frac{\pi}{2}$

= 4 $\frac{\pi}{2}$

= 2 $\pi$
6. The curve is symmetric about the x-axis, distance between the axis of symmetry and the maximum point is 1 unit.
Therefore, amplitude = 1 unit.

Graphing Trig Function y = Cos x.

Let us assume the values for x and find the corresponding values for y.
The following table shows the values of y for the corresponding values of y.
x
- 2 $\pi$
-3 $\frac{\pi}{2}$ - $\pi$
- $\frac{\pi}{2}$ 0
$\frac{\pi}{2}$ $\pi$ 3 $\frac{\pi}{2}$ 2 $\pi$
y 1
0
-1
0
1
0
-1
0
1

cosx
From the above graph we observe that
1. The curve is continuous from - 2 $\pi$ to + 2$\pi$
2. The curve does not pass through the origin.
3. The domain consists of all real numbers.
In the above graph Domain = { x : x $\epsilon$ [ - 2 $\pi$, 2 $\pi$ ] }
4. The graph has its maximum value y = 1 and the min value y = -1.
Therefore, the range of the function is { x : x $\epsilon$ [-1, 1 ], x is a real number }
5. The curve reaches maximum at x = - 2 $\frac{\pi}{2}$ and x = 0
therefore the period of the function is = 0 - ( - 2 $\pi$ )
= 2 $\pi$
= 2 $\pi$

6. The curve is symmetric about the x-axis, distance between the axis of symmetry and the maximum point is 1 unit.
Therefore, amplitude = 1 unit.

Graphing Trig function y = Tan x.

Let us assume the values for x and find the corresponding values for y.
The following table shows the values of y for the corresponding values of y.
tanx
From the above graph we observe that
1. The curve is continuous from - $\frac{\pi}{2}$ to + $\frac{\pi}{2}$ and is not continuous at all real numbers.
2. The graph does not touch the line x = n $\frac{\pi}{2}$ where n is an integer. Hence the function is not defined when x = n $\frac{\pi}{2}$.
2. The curve passes through the origin.
3. The domain consists of real numbers between -$\frac{\pi}{2}$ and $\frac{\pi}{2}$..

In the above graph Domain = (- 3$\frac{\pi}{2}$, - $\frac{\pi}{2}$) , (-$\frac{\pi}{2}$, $\frac{\pi}{2}$) , ($\frac{\pi}{2}$, 3$\frac{\pi}{2}$),

4. The graph has its maximum value which is + $\infty$ and the minimum value which is - $\infty$.
Therefore, the range of the function is { x : x $\epsilon$ ( -$\infty$, $\infty$ ) }

5. The curve reaches infinity at x = -$\frac{\pi}{2}$ and $\frac{\pi}{2}$,

therefore the period of the function is = $\frac{\pi}{2}$ - ( -$\frac{\pi}{2}$)

= $\frac{\pi}{2}$ + $\frac{\pi}{2}$

= $\pi$
6. The curve is symmetric about the y-axis.

General Sine and Cosine Function:

The General form sine and cosine functions are written as

f ( x ) = a sin ( b ( x - c ) ) + d,
f ( x ) = a cos ( b ( x - c ) ) + d,
where ' a ' is the amplitude,
period is "2$\frac{\pi}{b}$"
the horizontal shift is c and
the vertical shift is d.

Let us see examples to graph the some of the trigonometric functions using the general function.

Solved Example

Question: y = 3 sin 2 (x - $\frac{\pi}{4}$) + 5
Solution:
 
We have y    = 3 Sin 2(x - $\frac{\pi}{4}$) + 5

We have the general form,
                          y     =  a sin ( bx + c ) + d,
Therefore, Amplitude      =  a = 3

Period of the function     = 2$\frac{\pi}{2}$

                        b          = $\pi$

The horizontal shift (c)  = $\frac{\pi}{4}$

Vertical shift       (d)     =  5
The graph of the above function will be as shown below.

Graphing Trig Function

Graphing Trigonometric Functions