Trigonometric multiple and sub-multiple angle formulas





Trigonometric multiple and sub-multiple angle formulas:

Prerequisite: Please consider studying following topics before you study this article for better grasp and understanding:

Trigonometric addition and subtraction formulas

Trigonometric Functions

Pythagorian Identities

 

In this tutorial we shall derive formula for trigonometric functions of multiple and sub-multiple angle  , For example:

\sin 2 \theta , \, \cos 3 \theta , \, \tan \frac{1}{2} \theta  etc.

Trigonometric multiple angle formulas:

Under the trigonometric multiple angle formulas we shall derive the formulas for double and triple and trigonometric formulas which are listed below:

Double Angle Formulas:

If In the trigonometric Addition and subtraction formulae  we put angle A=B then we can easily derive following double angle formulas:

\cos 2 A = \cos ^2 A - \sin ^2 A \\ or , \cos 2 A = 1 - \sin ^2 A - \sin ^2 A = 1 - 2 . \sin ^2 A \\ or , \cos 2 A = 1 - 2 ( 1 - \cos ^2 A) = 2 \cos ^2 A - 1 \\ \\ \sin 2 A = 2 \sin A \cos A \\ \\ \tan 2 A = \dfrac{2 \tan A}{1 - \tan ^2 A}

From the cosine double angle formula above ; we can also derive:

2 \sin ^2 A = 1 - \cos 2 A \\ \\ 2 \cos ^2 A = 1 + \cos 2 A

Triple Angle Formulas:

We shall now Derive the formulas for triple angle formulas for Sine and Cosine.

We know ,

\sin 3 \theta : \\ \\ = \sin ( 2 \theta + \theta ) \\ \\ = \sin \theta \cos 2 \theta + \cos \theta \ sin 2 \theta \\ \\ = \sin \theta ( 1 - 2 \sin ^2 \theta) + 2 ( 1 - \sin ^2 \theta ) \sin \theta \\ \\ = 3 \sin \theta - 4 \sin ^3 \theta

And :

\cos 3 \theta : \\ \\ = \cos ( 2 \theta + \theta) \\ \\ = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta \\ \\ = ( 2 \cos ^2 \theta - 1 ) \cos \theta - 2 \cos \theta ( 1 - \cos ^2 \theta) \\ \\ = 4 \cos ^3 \theta - 3 \cos \theta

Trigonometric Sub-Multiple angle formulas:

We shall now derive the trigonometric formulas for half angle formulas.

Half angle formulas:

By replacing A by \frac{1}{2} A In the double angle formulas above we can easily derive the following half angle formulas:

\cos A = \cos ^2 \frac{1}{2} A - \sin ^2 \frac{1}{2} A \\ \\ \sin A = 2 \sin \frac{1}{2} A \cos \frac{1}{2} A \\ \\ \tan A = \dfrac{2 \tan \frac{1}{2} A}{1 - \tan ^2 \frac{1}{2} A} \\ \\ \cos A = 2 \cos ^2 \frac{1}{2} A - 1 \\ \cos A = 1 - \sin ^2 \frac{1}{2} A \\ \\ 2 \sin ^2 \frac{1}{2} A = 1 - \cos A \\ \\ 2 \cos ^2 \frac{1}{2} A = 1 + \cos A

 



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