**Preliminaries**

- We were introduced to the trigonometric functions in class with their
**general**definitions for any angle . The specialty of these functions is that their values get repeated after an interval. This is because after every interval of , the angles have same initial and terminal arms. **Basics**: If we fix the initial arm of the angle as positive axis, with vertex at origin. Let be**ANY**point on the terminal arm of the angle. Let be . Then,

**Periodicity**: The characteristic of trigonometric functions, which repeat their values after a fixed interval, is known as**periodicity**. The smallest non-negative interval is known as the**fundamental period**.

**Trigonometric Ratios of Standard Angles**

### Main Content

**Equations and Solutions**

The value which satisfies an equation is known as a **solution**. For example, is an equation. On simplifying,

So, the solution is .

**Trigonometric Equations**

The equations involving trigonometric functions are known as trigonometric equations. The unknown values, which are to be found, represent the **measure of an angle**. For example, is satisfied by or .

**Principal Solutions**

The solution, which lies between and , is known as the **principal solution**.

**General Solutions**

We saw that the measure of full circle is radians or . Hence, after adding to above angle, we will still get a solution. So,

Similarly,

Thus, there are infinitely many solutions to above equations, apart from . These solutions are known as **general solutions**. Consider another example:

**Note**: We will have to use allied angles formulas and the formula sheet to solve problems of this kind, where we are asked to find the general solutions.

**Polar Coordinates**

The Cartesian coordinates and the polar coordinates are inter-convertible. See the figure below:

**Solving a Triangle**

A triangle has 3 sides, say, and 3 angles, . By solving a triangle, we mean, to find values of all lengths of sides and all angles of a triangle. See the figure below.

The side opposite to is denoted by and so on.

There are certain rules to be used to solve these problems. These are:

**Sine Rule**

is the radius of the circumcircle.

**Cosine Rule**

**Projection Rule**

**Half Angle Formulas**

is the semi-perimeter.

**Area of a Triangle**

The area of triangle is given by,

It is also given by

. This is known as **Heron’s formula**.

**Napier’s Analogies**

**Inverse Trigonometric Functions**

If a function is defined from set A to set B (which is one-one and onto), we can define an inverse function, from set B to set A. So, if , then and .

If , then . If , then .

Note that is different from . is , which is .

On the other hand, is the angle, whose sine is .

Corresponding to each of six trigonometric functions, we have 6 inverse trigonometric functions, i.e. .

**Recall **: Principal value is the value of angle, which lies between and .

**Properties of Inverse Trigonometric Functions**

**Set I**

**Set II**

**Set III**

**Set IV**

**Set V**

**Set VI**