De Moivre's Theorem and its Applications in Trigonometry

De Moivre's Theorem is a powerful tool in mathematics, particularly in the field of complex numbers. It states that for any real number x and any integer n, we have the following identity:

cos nx i sin nx (cos x i sin x)n

Expanding the Cosine and Sine Terms

Let's consider the expansion of De Moivre's Theorem for the specific case where n 3. We start by substituting n 3 into the theorem:

cos 3x i sin 3x (cos x i sin x)3

Expanding the right-hand side using the binomial theorem, we get:

(cos x i sin x)3 cos3x 3i cos2x sin x - 3 cos x sin2x - i sin3x

Separating the real and imaginary parts, we obtain:

cos 3x i sin 3x (cos3x - 3 cos x sin2x) i (3 cos2x sin x - sin3x)

By comparing the real and imaginary parts on both sides, we get the following trigonometric identities:

cos 3x 4 cos3x - 3 cos x

sin 3x 3 sin x - 4 sin3x

Deriving Trigonometric Identities

Another way to derive the identity for sin 3x without using De Moivre's Theorem directly is by using the angle addition formula for sine:

sin 3x sin (x 2x) sin x cos 2x cos x sin 2x

Using the double angle identities for sine and cosine, we have:

cos 2x cos2x - sin2x

sin 2x 2 sin x cos x

Substituting these into the equation for sin 3x, we get:

sin 3x sin x (cos2x - sin2x) cos x (2 sin x cos x)

Expanding and simplifying, we obtain:

sin 3x 3 sin x cos2x - sin3x

Further simplification using the Pythagorean identity, cos2x 1 - sin2x, yields:

sin 3x 3 sin x (1 - sin2x) - sin3x

sin 3x 3 sin x - 4 sin3x

Euler's Formula and Trigonometric Expressions

Another fascinating way to derive these identities is by using Euler's formula, which expresses the sine function in terms of the exponential function:

sin x (eix - e-ix) / (2i)

Applying this to sin 3x, we get:

sin 3x (e3ix - e-3ix) / (2i)

Using the binomial theorem, we get:

(eix - e-ix)3 e3ix - 3 eix e-ix 3 e-ix eix - e-3ix

Reducing this, we have:

sin 3x (1/2i) * (e3ix - e-3ix) - 3/2 (eix - e-ix) / (2i)

Simplifying, we obtain:

sin 3x (3/4) sin x - (1/4) sin 3x

Thus, rearranging, we get:

sin 3x 3 sin x - 4 sin3x