Problem 40 - A First Course in Abstract Algebra

Question

a. Dertive 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

The formula is \(\cos 3\theta = 4\cos^3\theta - 3\cos\theta\).

Step 1: Represent \\(\cos 3\theta\\) using Euler's Formula

Start with Euler's formula: \(e^{i\theta} = \cos \theta + i\sin \theta\). Therefore, \(e^{i3\theta} = (e^{i\theta})^3.\)Expanding the right side using the cube identity gives: \[e^{i3\theta} = (\cos \theta + i \sin \theta)^3.\]Use binomial expansion: \[= \cos^3 \theta + 3i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta.\]

Step 2: Simplify to Real and Imaginary Parts

Separate the real and imaginary parts of the expression to isolate \(\cos 3\theta\) and \(\sin 3\theta\):\[\cos 3\theta + i \sin 3\theta = \cos^3 \theta - 3 \cos \theta \sin^2 \theta + i(3 \cos^2 \theta \sin \theta - \sin^3 \theta).\]Thus, the real part equates to \(\cos 3\theta\):\[\cos 3\theta = \cos^3 \theta - 3 \cos \theta \sin^2 \theta.\]

Step 3: Use \\(\sin^2 \theta + \cos^2 \theta = 1\\) to Simplify

Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to replace \(\sin^2 \theta\) with \(1 - \cos^2 \theta\):\[\cos 3\theta = \cos^3 \theta - 3 \cos \theta (1 - \cos^2 \theta).\]

Step 4: Distribute and Simplify Further

Distribute \(-3\cos\theta\):\[\cos 3\theta = \cos^3 \theta - 3 \cos \theta + 3 \cos^3 \theta\]Combine like terms:\[\cos 3\theta = 4\cos^3 \theta - 3\cos \theta.\]

Step 5: Final Step: Verify the Derived Expression with Known Formula

Verify that this expression matches the standard identity for triple angle:\[\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta.\]This confirms the derivation using Euler's Formula and trigonometric identities is correct.

Key Concepts

Trigonometric Identities

In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all value inputs to those functions. The most fundamental trigonometric identities are those that relate the sine and cosine of any angle \( \theta \). A classic and deeply important one is the Pythagorean identity:

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

This identity underpins a lot of work in trigonometry, allowing us to express one trigonometric function in terms of another. It's particularly useful when simplifying expressions and solving equations in trigonometry.
In the context of deriving \( \cos 3\theta \), this identity helps to substitute and eliminate \( \sin^2 \theta \), turning it into something more manageable like \( 1 - \cos^2 \theta \). Doing so simplifies the calculation and provides a clearer path to the solution.

Triple Angle Formula

The triple angle formulas are specific cases of angle identities used in trigonometry, especially to express a trigonometric function of an angle in terms of functions of multiple angles. One such formula for cosine is:

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

This formula is derived from the identity for \( \cos(\alpha + \beta + \gamma) \) and helps in expressing \( \cos \) of triple angles in purely polynomial terms of single angle \( \cos \) functions.
In exercises like deriving \( \cos 3\theta \) in terms of \( \cos \theta \) and \( \sin \theta \), the triple angle formula can reaffirm consistency between calculated expressions and known trigonometric identities.

Binomial Expansion

The binomial expansion is a formula used to expand expressions that are raised to a power, in the format \( (x + y)^n \). In relation to Euler's formula and solving trigonometric expressions:

• When \((\cos \theta + i\sin \theta)^3\) expands, each term follows from the binomial theorem, leading to terms like \( \cos^3 \theta + 3i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta \).

The process relies on expanding into real and imaginary components, keeping track of \(i\), the imaginary unit, to combine terms appropriately.
This allows expression of complex angles in terms of easier-to-manage base trigonometric functions, facilitating further simplification using known identities. Binomial expansion is particularly potent for expressions involving powers of sums in algebra and complex numbers.

Real and Imaginary Parts Separation

When dealing with Euler's formula and complex expressions, separating the real and imaginary parts is a key step to finding solutions. This technique separates the terms into components that contain \(i\) (imaginary parts) and those that don't (real parts).

• For instance, from \(e^{i3\theta} = \cos^3 \theta - 3 \cos \theta \sin^2 \theta + i(3 \cos^2 \theta \sin \theta - \sin^3 \theta)\), the real part, \( \cos 3\theta \), is distinctly separated from the imaginary part, \( \sin 3\theta \).

Real and imaginary separation is essential when working with complex numbers and applications where understanding of different components allows deeper insights.
By isolating these parts, one can apply identities and resolve terms separately, ensuring accuracy in outcomes when handling complex trigonometric equations.