Expanding sin3x and cos3x: A Comprehensive Guide

In this article, we delve into the trigonometric identities that allow us to expand sin3x and cos3x based on their underlying angles. Understanding these expansions is crucial for solving trigonometric equations and simplifying trigonometric expressions.

Understanding the Basic Angle Addition Formulas

Before diving into the expansions, it's important to revisit the basic angle addition formulas for sin and cos. These formulas allow us to express trigonometric functions of sums of angles in terms of the trigonometric functions of the individual angles.

Expanding sin3x

The expansion of sin3x is derived using the triple angle identity:

sin3x 3sinx - 4sin3x

Expanding cos3x

Similarly, the expansion of cos3x is given by the following triple angle identity:

cos3x 4cos3x - 3cosx

Applying de Moivre's Theorem

Another method to derive these expansions involves the use of de Moivre's Theorem, which states that for any real number x and any integer n:

eix cosx isinx

Raising both sides to the power of 3, we get:

e3ix cos3x isin3x

Substituting the series expansion of e^ix on the left side, we have:

e3ix (1 3ix - (3ix)2 / 2! (3ix)3 / 3! - ...)

Expanding and simplifying, we obtain:

cos3x isin3x cos3x - 3cosxsin2x i(3cos2xsinx - sin3x)

Comparing the real and imaginary parts, we get:

cos3x cos3x - 3cosxsin2x

sin3x 3cos2xsinx - sin3x

Alternative Derivation Using Trigonometric Identities

We can also derive these identities using the trigonometric identities:

sin(a b) sina cosb cosa sinb

cos(a b) cosa cosb - sina sinb

sin2x 2sina cosx

cos2x cos2x - sin2x

sin2x cos2x 1

Applying these identities, we can derive:

sin3x  3sinx - 4sin3xcos3x  4cos3x - 3cosx

Summary

In summary, the expansions of sin3x and cos3x are: