- 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
- 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,
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