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.
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