Question

Use deMoivre's Theorem to write \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta .\)

Short answer

The expression \( \sin^3 \theta \) can be written in terms of \( \sin \theta \) and \( \sin 3\theta \) as \( \frac{1}{4}(\sin 3\theta + 3\sin \theta) \).

Step-by-step solution

Express using deMoivre's Theorem

Express \( \sin^3 \theta \) as \( \frac{(e^{i\theta} - e^{-i\theta})^3}{(2i)^3} \) using the complex form of sine function and apply Binomial Theorem.

Simplify the equation

Simplify the equation and convert the equation in the form of cosine function. After simplifying, we get \( \frac{e^{3i\theta} + 3e^{i\theta} - 3e^{-i\theta} - e^{-3i\theta}}{8i} \).

Apply trigonometric identities

Now apply trigonometric identities to transform the equation back into sinusoidal form. The expression simplifies to \( \frac{1}{4}(\sin 3\theta + 3\sin \theta) \).

Key concepts

Complex Numbers

Complex numbers might sound complicated, but they're just numbers that have a real part and an imaginary part. We write them as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary part is given by \( bi \), with \( i \) being the imaginary unit. One of the unique features of complex numbers is that they can be represented in both rectangular form (like \( a + bi \)) and polar form. Using the polar form, a complex number is expressed as \( re^{i\theta} = r(\cos \theta + i\sin \theta) \). Here, \( r \) is the magnitude of the complex number, and \( \theta \) is called the argument, which reflects the angle formed with the positive real axis.

In mathematics, complex numbers can cleanly express oscillations and rotations, which makes them perfect tools for problems related to waves and periodic functions, like sine and cosine. Using this representation, De Moivre's Theorem comes into play, allowing us to easily handle powers and roots of complex numbers. De Moivre's Theorem states that for any real number \( \theta \) and integer \( n \), \( (\cos \theta + i \sin \theta)^n = \cos(n \theta) + i \sin(n \theta) \). This allows us to convert complex expressions involving powers into trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. They're essential for simplifying expressions involving trigonometric functions like sine, cosine, and tangent. These identities help in transforming complex trigonometric expressions into simpler or more useful forms, which is especially important in calculus and physics.

Some common trigonometric identities include the Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), and angle sum identities like \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \). In the context of De Moivre's Theorem and the exercise, identities help us turn complex exponential expressions back into sine or cosine terms.

For example, using De Moivre’s Theorem, you can break down \( e^{3i\theta} \) into its trigonometric sum: \( \cos(3\theta) + i\sin(3\theta) \). Similarly, applying sine and cosine sum identities allows us to ultimately convert expressions like \( \sin^3 \theta \) back using simpler trigonometric forms, such as \( \frac{1}{4} (\sin 3\theta + 3\sin \theta) \), which aligns with sinusoidal patterns found in wave functions.

Binomial Theorem

The Binomial Theorem is a fundamental principle that describes the algebraic expansion of powers of a binomial. In simple terms, it shows how to expand expressions raised to a power, such as \( (x + y)^n \). The theorem states: \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\] where \( \binom{n}{k} \) represents the binomial coefficients.

These coefficients are the numbers you see in Pascal's triangle and can be calculated with \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This theorem is highly valuable in both algebra and calculus because it simplifies computations by breaking them into manageable parts.

In the exercise, De Moivre's Theorem is applied in conjunction with the Binomial Theorem to expand and simplify the expression \( (e^{i\theta} - e^{-i\theta})^3 \), ultimately expressing \( \sin^3 \theta \) in terms of \( \sin \theta \) and \( \sin 3\theta \). By recognizing and simplifying this binomial expression, we calculate the trigonometric identities for sine, thus easing the method to solve trigonometric equations.