Proving the Trigonometric Identity: sin3x 3sinx - 4sin^3x

Trigonometry is a fascinating branch of mathematics that finds applications in various fields such as physics, engineering, and computer science. One of the essential concepts in trigonometry is the identity sin3x 3sinx - 4sin^3x. This identity can be demonstrated in several ways, including using complex numbers and De Moivre’s Theorem. Let’s explore these methods in detail.

Using Complex Numbers and De Moivre’s Theorem

De Moivre’s Theorem states that for any real number ( x ) and integer ( n ), the complex number ( (cos x i sin x)^n ) can be expressed as ( cos(nx) i sin(nx) ). We can apply this theorem to prove the identity in question.

Step 1: Applying De Moivre’s Theorem to cos3x and sin3x

From De Moivre’s Theorem:

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

Expanding the left-hand side using the binomial theorem:

( (cos x i sin x)^3 cos^3 x 3i cos^2 x sin x - 3 cos x sin^2 x - i sin^3 x )

Separating the real and imaginary parts, we get:

( cos 3x i sin 3x (cos^3 x - 3 cos x sin^2 x) i (3 cos^2 x sin x - sin^3 x) )

Matching the imaginary parts on both sides:

( sin 3x 3 cos^2 x sin x - sin^3 x )

Using the identity ( cos^2 x 1 - sin^2 x ), substitute and simplify:

( sin 3x 3 (1 - sin^2 x) sin x - sin^3 x )

( sin 3x 3 sin x - 3 sin^3 x - sin^3 x )

( sin 3x 3 sin x - 4 sin^3 x )

Alternative Methods for Proving the Identity

Graphical Method Using Desmos

Desmos, a powerful graphing calculator, can visually confirm the identity. By plotting the functions on Desmos, you can observe that they are indeed the same. Although Desmos does not directly calculate trigonometric functions to the power of three, you can use squared terms and cube roots to verify the identity.

Using Basic Trigonometric Identities

Another way to prove the identity involves using basic trigonometric identities. Start by expressing ( sin 3x ) in terms of ( sin x ) and ( cos x ).

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

Using the double-angle identities:

( sin 2x 2 sin x cos x )

( cos 2x cos^2 x - sin^2 x )

Substitute these into the equation:

( sin 3x 2 sin x cos x cos x (cos^2 x - sin^2 x) sin x )

( sin 3x 2 sin x cos^2 x cos^2 x sin x - sin^3 x )

Combine like terms:

( sin 3x 3 sin x cos^2 x - sin^3 x )

Substitute ( cos^2 x 1 - sin^2 x ) again:

( sin 3x 3 sin x (1 - sin^2 x) - sin^3 x )

( sin 3x 3 sin x - 3 sin^3 x - sin^3 x )

( sin 3x 3 sin x - 4 sin^3 x )

Solving for x

To solve for ( x ) given the equation ( sin 3x 3 sin x - 4 sin^3 x ), set the equation to zero:

( 3 sin x - 4 sin^3 x - sin 3x 0 )

This simplifies to:

( 0 0 )

Thus, the identity is true for all values of ( x ).

Conclusion

Proving the trigonometric identity ( sin 3x 3 sin x - 4 sin^3 x ) involves several methods including the use of complex numbers, De Moivre’s Theorem, and basic trigonometric identities. Each method provides a unique perspective on the identity, reinforcing its validity and deepening our understanding of trigonometry. By exploring these methods, you can enhance your problem-solving skills in trigonometry and strengthen your foundation in this essential area of mathematics.