The trigonometric functions are those which are real numbers found by finding the ratio of the pair of sides of a right angled triangle formed by the corresponding angles of a triangle.

These trigonometric functions are useful to solve any type of triangles for the angles and sides, to find the distance between two ships or two aeroplanes, to calculate the velocity of the bullet fired from a gun etc. The graph of the trigonometric function help us to find the max and min of a function and also to find the monotonicity of the function. The modelling of blood pressure, rotation of giant wheel, the life cycle of a star are all sinusoidal function, which will help us to predict the value of the function at a particular value of y.

The derivative of a trigonometric function will help us to find the slope of a curve at a particular point on the graph and the integration of the trigonometric function will help us to find the area bounded by the curve between any two values of x. In this section let us discuss about the trigonometric functions, trigonometric functions graphs, derivative of trigonometric functions and the integrals of trigonometric functions and the trigonometric functions problems.

The trigonometric functions of an angle are defined as the ratio of the corresponding pairs of sides of a right angled triangle.

Six Trigonometric Functions:

Six Trigonometric Functions
From the above triangle, the six trigonometric ratios are written as follows.

1. Sin ( $\theta$ ) = $\frac{Opposite\;Side}{Hypotenuse}$ = $\frac{a}{c}$

2. Cos $\theta$ = $\frac{Adjacent \;Side}{Hypotenuse}$ = $\frac{b}{c}$

3. Tan $\theta$ = $\frac{Opposite\;Side}{Adjacent\;Side}$ = $\frac{a}{b}$

4. Csc $\theta$ = $\frac{1}{sin\;\theta}$ = $\frac{Hypotenuse}{Opposite\;Side}$ = $\frac{c}{a}$

5. Sec $\theta$ = $\frac{1}{cos\;\theta}$ = $\frac{Hypotenuse}{Adjacent \;Side}$ = $\frac{c}{b}$

6. Cot $\theta$ = $\frac{1}{\tan \theta}$ = $\frac{Adjacent\;Side}{Opposite\;Side}$ = $\frac{b}{a}$

Properties of Trigonometric Functions:

When we draw a unit circle on the two-dimensional graph, the trigonometric functions change their signs according to the position of the angle in different quadrants.

Let P(x, y) be a point on the unit circle.

Then x = cos $\theta$ and y = sin $\theta$.

First and Second Quadrant

Third and Fourth Quadrant

The above diagram shows the angle in the four different quadrants.

The following table help us to find the sign of the trigonometric functions in different quadrants.
Quadrants
Trigonometric Functions
First Quadrant
Second Quadrant
Third Quadrant
Fourth Quadrant
Sin $\theta$
Positive Positive Negative Negative
Csc $\theta$
Positive Positive Negative Negative
Cos $\theta$ Positive Negative Negative Positive
Sec $\theta$ Positive Negative Negative Positive
Tan $\theta$ Positive Negative Positive Negative
Cot $\theta$ Positive Negative Positive Negative

While graphing Trigonometric functions we should know the periods of each function.
The periods of sine and cosine functions are 2 $\pi$ where as the periods of tangent function is $\pi$ Similarly the reciprocal functions will also have the same periods as per the original (basic) functions.

The general form of a trigonometric function is
y = a Sin (bx+c) + d,
y = a Cos (bx+c) + d
where a is the amplitude,
period = 2 $\frac{\pi}{b}$

Horizontal Shift = -$\frac{c}{b}$

Vertical shift (Axis of symmetry) y = d

Solved Example

Question: Graph the function y = 5 cos ($\frac{1}{2}$ x - $\frac{\pi}{4}$) + 3
Solution:
 
The Given function is
                                                   y = 5 cos ($\frac{1}{2}$ x - $\frac{\pi}{4}$) + 3
We have the general form,
                                                  y = a Cos ( bx+c) + d
The given equation can be written in the form,
                                                  y = 5 cos [ $\frac{1}{2}$ (x - $\frac{\pi}{2}$) ] + 3
comparing with the general form, we get
                               amplitude (a) = 5
                                                  b = $\frac{1}{2}$
                          Axis of symmetry = y = 3
The graph will be as follows.
Cosine Graph
 

The following table shows the derivatives of trigonometric functions.

Serial Number
Function f ( x)
Derivative of the function f ' ( x )
1 sin x
cos x
2 cos x
- sin x
3 tan x sec2 x
4 csc x
- csc x. cot x
5 sec x
sec x. tan x
6 cot x - csc2 x
7 sin ax a cos ax
8 cos ax
- a sin ax
9 tan ax a sec2 ax
10 csc ax
- a csc ax. cot ax
11 sec ax
a sec ax . tan ax
12 cot ax
- a csc2 ax

The following table shows the integrals of trigonometric functions.
Serial Number
Function f (x)
Integral of the function $\int f(x)dx$
1 sin x - cos x + C
2 cos x
sin x + C
3 tan x
- ln | cos x | + C
ln | sec x | + C
4 csc x
- ln | csc x + cot x | + C
5 sec x
ln | sec x + tan x | + C
6 cot x
ln | sin x | + C
7 sin ax
- $\frac{1}{a}$ cos ax + C
8 cos ax
$\frac{1}{a}$ sin ax + C
9 tan ax
- $\frac{1}{a}$ ln | cos ax | + C

$\frac{1}{a}$ ln | sec ax | + C
10 csc ax
- $\frac{1}{a}$ ln | csc ax | cot ax | + C
11 sec ax
$\frac{1}{a}$ ln | sec ax + tan ax | + C
12 cot ax
- $\frac{1}{a}$ csc2 ax + C


Solved Example

Question: John was riding on a Ferris wheel. John's height above the ground in terms of time is represented by the graph as shown below.Ferri's Wheel

1. What is the period of this function?
2. Find the amplitude.
3. Find the range of the function.
4. Find the equation of the axis of symmetry.
5. What will be his height above the ground at 25th second.
6. At what time he reaches a height of 6 m between 15th and 25th sec.
Solution:
 
1. Period of the function:
   The graph reaches maximum when x = 2 sec and x = 22 sec.
Therefore, period of the function = 22 - 2 = 20 sec
2. Amplitude: The maximum point of the graph is y = 7 and the minimum point is y = 1

   Amplitude is $\frac{(7-1)}{2}$ = $\frac{6}{2}$ = 3

3. Range: The curve is between y = 1 and y = 7.
Therefore, the range of the function is [ 1, 7 ].
4. Equation of Axis of symmetry:

    y = $\frac{Maximum\;+\;Minimum}{2}$ = $\frac{1\;+\;7}{2}$ = $\frac{8}{2}$ = 4

   Equation of axis of symmetry is y = 4
5. At 25th second, John will be at a height of 5 m above the ground.
6. at 20th second,  John will be at a height of 6 m above the ground.