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.

## Six Trigonometric Functions

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:

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 $\theta$.

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.
 QuadrantsTrigonometric 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

### Inverse Trigonometric Functions

 Sine Cosine Tangent Sec Csc Cot Law of Sines and Cosines Law of Tangents
 Continuity of Trigonometric Functions Derivative of Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Graphs of Trigonometric Functions Hyperbolic Trigonometric Functions Trigonometric Functions on the Unit Circle Trigonometric Equation Inverse Trigonometric Ratios Pythagorean Trigonometric Identities Trigonometry Formulas Right Angle Trigonometry Trigonometry Half Angle Formula
 Formulas for Trigonometric Functions Inverse Trigonometric Functions Calculator Free Trigonometric Identities Solver