Problem 40 - A First Course in Abstract Algebra

Question

a. Dertve a formula for \(\cos 3 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\) using Euler's formula. b. Derive the formula \(\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta\) from part (a) and the identity \(\sin ^{2} \theta+\cos ^{2} \theta=1\). (We will have use for this identity in Section 32.)

Step-by-Step Solution

Answer

\(\cos 3\theta = 4 \cos^3 \theta - 3\cos \theta\).

Step 1: Express Euler's Formula

Euler's formula states that any complex exponential can be expressed as the sum of sine and cosine functions: \( e^{i\theta} = \cos \theta + i \sin \theta \). We can use this to express \(e^{i(3\theta)}\) as \((e^{i\theta})^3\).

Step 2: Expand \( (e^{i\theta})^3 \) using Binomial Theorem

Calculate \((e^{i\theta})^3 = (\cos \theta + i \sin \theta)^3\) by expanding it using the binomial theorem: \(a^3 + 3a^2b + 3ab^2 + b^3\), where \(a = \cos \theta\) and \(b = i \sin \theta\).

Step 3: Simplify \((\cos \theta + i \sin \theta)^3\)

Apply the binomial expansion to get: \((\cos \theta)^3 + 3(\cos \theta)^2(i \sin \theta) + 3(\cos \theta)(i \sin \theta)^2 + (i \sin \theta)^3\). Simplify the powers of \(i\): \(i^2 = -1\) and \(i^3 = -i\).

Step 4: Separate Real and Imaginary Components

After simplifying, collect the real part (cosine terms) and imaginary part (sine terms): \( \cos 3\theta + i \sin 3\theta = \cos^3 \theta - 3\cos \theta \sin^2 \theta + i(3\cos^2 \theta \sin \theta - \sin^3 \theta) \).

Step 5: Isolate \(\cos 3\theta\)

Identify the real component from the expression obtained in Step 4 as \(\cos 3\theta = \cos^3 \theta - 3\cos \theta \sin^2 \theta\).

Step 6: Substitute Identity \(\sin^2 \theta = 1 - \cos^2 \theta\)

Use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to substitute in \(\cos 3\theta = \cos^3 \theta - 3\cos \theta (1 - \cos^2 \theta)\).

Step 7: Simplify the Expression

Distribute the \(-3\cos \theta\) and simplify: \( \cos 3\theta = \cos^3 \theta - 3\cos \theta + 3\cos^3 \theta\). Combine like terms to derive: \(\cos 3\theta = 4\cos^3 \theta - 3\cos \theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are particularly useful tools in mathematics, helping simplify and solve equations involving angles. At the core, these identities relate the angles and sides of triangles, providing a means to express trigonometric functions such as sine, cosine, and tangent in various forms. One of the most fundamental trigonometric identities is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is crucial when simplifying expressions, as seen in our Euler's formula exercise. It allows re-expression of sine terms in terms of cosine and vice versa. For example, if you're given \( \sin^2 \theta \), you can express it as \( 1 - \cos^2 \theta \), simplifying complex trigonometric expressions significantly. Another useful identity, the double angle identity for cosine, states:

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• It can also be rewritten in terms of either sine or cosine alone, like \( 2\cos^2 \theta - 1 \) or \( 1 - 2\sin^2 \theta \).

Applying these identities makes manipulating expressions more manageable, especially in conjunction with other mathematical theorems like Euler's formula.

Binomial Theorem

The binomial theorem is a powerful mathematical tool used to expand expressions that are raised to a power, like \( (a + b)^n \). This theorem states:

• \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

Where \( \binom{n}{k} \) represents the binomial coefficients, which can be calculated using combinations. This theorem becomes extremely useful for expanding powers of sums involving complex numbers, as demonstrated in the Euler's formula exercise. In our example, the expression \( (\cos \theta + i \sin \theta)^3 \) can be expanded using the binomial theorem. It leads to:

• \( \cos^3 \theta + 3\cos^2 \theta(i \sin \theta) + 3\cos \theta(i \sin \theta)^2 + (i \sin \theta)^3 \)

By expanding like this, we can separate the real and imaginary components more effectively, which is essential for forms derived via Euler's method. The binomial theorem, therefore, provides a structured way to handle and simplify expressions that could otherwise be quite cumbersome.

Complex Numbers

Complex numbers are a vast and interesting topic in mathematics, expanding beyond the limitations of real numbers. Represented typically as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit (\( i^2 = -1 \)), complex numbers allow for the solving of equations that don't have solutions among the reals. Euler's formula itself (

• \( e^{i\theta} = \cos \theta + i \sin \theta \)

) exploits the properties of complex numbers. This relationship elegantly connects exponential functions with trigonometric functions, providing a powerful tool for representing rotations and oscillations. When dealing with complex number expressions like \( (\cos \theta + i \sin \theta)^n \), deploying complex algebra and trigonometric identities can elucidate the imaginary and real components clearly. By understanding the fundamental operations like multiplying complex numbers and simplifying, one can handle more intricate mathematical challenges as shown in deriving trigonometric identities using complex numbers via Euler's formula. With complex numbers, the world of mathematics offers a deeper and more interconnected view of equations and their solutions, broadening analytic techniques significantly.