Trigonometric Functions-II, Objectives, Addition and Multiplication of Trigonometric Functions, Addition Formula, Corollary 1

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  • In the previous lesson, you have learnt trigonometric functions of real numbers, drawn and interpreted the graphs of trigonometric functions.

  • In this lesson we will establish addition and subtraction formulae for , and .

  • We will also state the formulae for the multiple and sub multiples of angles and solve examples thereof.

  • The general solutions of simple trigonometric functions will also be discussed in the lesson.


  • After studying this lesson, you will be able to:

  • Write trigonometric functions of where , are real nunbers;

  • Establish the addition and subtraction formulae for :



  • Solve problems using the addition and subtraction formulae;

  • State the formulae for the multiples and sub-multiples of angles such as , , , , ,,, and;

  • Solve simple trigonometric equations of the type:

, ,

Addition and Multiplication of Trigonometric Functions

  • In earlier sections we have learnt about circular measure of angles, trigonometric functions, values of trigonometric functions of specific numbers and of allied numbers.

  • You may now be interested to know whether with the given values of trigonometric functions of any two numbers and , it is possible to find trigonometric functions of sums or differences.

  • You will see how trigonometric functions of sum or difference of numbers are connected with those of individual numbers. This will help you, for instance, to find the value of trigonometric functions of and etc.

can be expressed as and can be expressed as

Addition Formula

Addition Formula

Addition Formula

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For any two numbers and ,

In given figure trace out

Where points , , , lie on the unit circle.

Coordinates of , , , will be ,


and .

From the given figure, we have side side, (each angle ), sideside (by )



Corollary 1

For any two numbers and ,


Replace by in


Find the value of each of the following: