Question

Consider the complex number \(z=e^{i \theta}=\cos \theta+i \sin \theta .\) (a) By evaluating \(z^{2}\) two different ways, prove the trig identities \(\cos 2 \theta=\cos ^{2} \thetaQEDQED-\sin ^{2} \theta\) and \(\sin 2 \theta=2 \sin \theta \cos \theta .\) (b) Use the same technique to find corresponding identities for \(\cos 3 \theta\) and \(\sin 3 \theta\).

Short answer

Use Euler's formula and equate real/imaginary components.

Step-by-step solution

Evaluate \( z^2 \) directly from exponential form

Since \( z = e^{i\theta} \), we raise it to the power of 2: \( z^2 = (e^{i\theta})^2 = e^{2i\theta} \). By Euler's formula, we know \( e^{2i\theta} = \cos(2\theta) + i\sin(2\theta) \).

Evaluate \( z^2 \) using trigonometric form

Given that \( z = \cos \theta + i \sin \theta \), we can calculate \( z^2 = (\cos \theta + i \sin \theta)^2 \). Using the identity \((a+b)^2 = a^2 + 2ab + b^2\), we expand it: \[ z^2 = \cos^2 \theta + 2i \cos \theta \sin \theta - \sin^2 \theta \].

Equate real parts to find \( \cos 2\theta \)

The real part of \( z^2 = \cos^2 \theta - \sin^2 \theta \) from the second method should equal the real part from Step 1, \( \cos 2\theta \). Thus, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).

Equate imaginary parts to find \( \sin 2\theta \)

The imaginary part of \( z^2 \) is \( 2i \cos \theta \sin \theta \) from the trigonometric form, which equals the imaginary part from Step 1, \( i \sin 2\theta \). Therefore, \( \sin 2\theta = 2 \sin \theta \cos \theta \).

Find \( z^3 \) using exponential form

For part (b), consider \( z^3 = (e^{i\theta})^3 = e^{3i\theta} \). By Euler’s formula, \( e^{3i\theta} = \cos(3\theta) + i\sin(3\theta) \).

Find \( z^3 \) using trigonometric expansion

We can expand \( z^3 = (\cos \theta + i \sin \theta)^3 \) using the binomial theorem. Calculate the terms: \[ z^3 = \cos^3 \theta + 3i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta \].

Simplify and equate real parts for \( \cos 3\theta \)

Group the real parts: \( \cos^3 \theta - 3\cos \theta \sin^2 \theta \). Equating it to the real part from Step 5, \( \cos 3\theta \): \( \cos 3\theta = \cos^3 \theta - 3\cos \theta \sin^2 \theta \).

Simplify and equate imaginary parts for \( \sin 3\theta \)

Group the imaginary parts: \( 3\cos^2 \theta \sin \theta - \sin^3 \theta \) and equate it to the imaginary part from Step 5, \( \sin 3\theta \): \( \sin 3\theta = 3\cos^2 \theta \sin \theta - \sin^3 \theta \).

Key concepts

Complex Numbers

Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). The real part is \( a \) and the imaginary part is \( b \). Complex numbers extend the idea of one-dimensional real numbers to two-dimensional numbers by representing them on a plane.

The complex plane, also known as the Argand plane, is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers. Each point on this plane corresponds to a complex number. The formula \( z = e^{i\theta} = \cos \theta + i \sin \theta \), used in the exercise above, is a well-known representation of complex numbers that connects them to trigonometric functions.

A crucial aspect of complex numbers is their modulus and argument. The modulus is the distance from the origin to the point in the complex plane and is calculated via \( |z| = \sqrt{a^2 + b^2} \). The argument is the angle formed with the positive real axis, often denoted as \( \theta \), calculated using trigonometric functions. Complex numbers provide a powerful toolset for solving equations that do not have real solutions and play a critical role in advanced mathematics and engineering.

Euler's Formula

Euler's formula is one of the most remarkable equations in mathematics. It states that for any real number \( \theta \), \( e^{i\theta} = \cos \theta + i \sin \theta \). This formula elegantly bridges the fields of trigonometry and complex analysis.

The formula provides a way to express complex numbers in polar form, making calculations involving complex numbers more manageable. Specifically, through Euler's formula, the exponential function \( e^{i\theta} \) links exponential growth to trigonometric oscillation, describing how complex numbers rotate around the origin in the complex plane.

Using Euler's formula, we can derive several trigonometric identities, as demonstrated in the exercise with \( z^{2} = e^{2i\theta} \), leading to the identities \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) and \( \sin 2\theta = 2\sin \theta \cos \theta \). This not only shows how trigonometric identities can be derived from complex exponentials but highlights the utility of Euler’s formula in simplifying complex calculations.

This blending of trigonometry and complex numbers simplifies many mathematical problems, offering insights into the behavior of functions and phenomena that are inherently periodic or oscillatory.

Binomial Theorem

The binomial theorem provides a formula for expanding powers of sums. Specifically, it states that \((a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\), where \( \binom{n}{k} \) is a binomial coefficient. In our exercise, this theorem allows us to expand \((\cos \theta + i \sin \theta)^n\) to find expressions for higher powers of complex numbers.

When applying the binomial theorem to trigonometric forms of complex numbers, each term in the expansion represents a combination of real and imaginary parts raised to different powers. The theorem incorporates the idea of permutations of terms and coefficients, allowing a structured way to expand polynomials.

In the context of this exercise, using the binomial theorem helps derive formulas for \( \cos 3\theta \) and \( \sin 3\theta \). By expanding \( (\cos \theta + i \sin \theta)^3 \) and separating real and imaginary components, we obtain new trigonometric identities.

The binomial theorem is a powerful computational tool, not only facilitating the multiplication of expressions but also giving rise to identities across various branches of mathematics.