Problem 38 - Advanced Engineering Mathematics

Question

Use the result \((\cos \theta+i \sin \theta)^{3}=\cos 3 \theta+i \sin 3 \theta\) to find trigonometric identities for \(\cos 3 \theta\) and \(\sin 3 \theta\).

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Given Expression

We start with the expression \((\cos \theta + i \sin \theta)^3\) and recognize it as a complex number expressed in polar form. According to De Moivre's Theorem, raising a complex number in the form \((\cos \theta + i \sin \theta)^n\) gives us \(\cos(n\theta) + i \sin(n\theta)\). Here, \(n = 3\).

Step 2: Expand the Expression Using Binomial Theorem

Expand \((\cos \theta + i \sin \theta)^3\) using the binomial theorem: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Substitute \(a = \cos \theta\) and \(b = i \sin \theta\):\[ (\cos \theta + i \sin \theta)^3 = \cos^3 \theta + 3\cos^2 \theta (i \sin \theta) + 3\cos \theta (i \sin \theta)^2 + (i \sin \theta)^3 \]

Step 3: Simplify the Expression

Simplify each term. The first term is \(\cos^3 \theta\). The second term is \(3 \cos^2 \theta \cdot i \sin \theta = 3 i \cos^2 \theta \sin \theta\). The third term is \(3 \cos \theta \cdot (i)^2(\sin^2 \theta) = -3 \cos \theta \sin^2 \theta\). The fourth term is \((i)^3(\sin \theta)^3 = -i \sin^3 \theta\). Therefore:\[ \cos^3 \theta - 3 \cos \theta \sin^2 \theta + i(3 \cos^2 \theta \sin \theta - \sin^3 \theta) \]

Step 4: Identify Real and Imaginary Parts

According to De Moivre's Theorem, and our expansion: The real part corresponds to \(\cos 3\theta\), and the imaginary part corresponds to \(\sin 3\theta\). Based on the expansion, identify:- Real Part: \(\cos 3\theta = \cos^3 \theta - 3 \cos \theta \sin^2 \theta\)- Imaginary Part: \(\sin 3\theta = 3 \cos^2 \theta \sin \theta - \sin^3 \theta\)

Step 5: Apply Double Angle Identities

Use \(\sin^2 \theta = 1 - \cos^2 \theta\) and \(\cos^2 \theta = 1 - \sin^2 \theta\) to further simplify the identities if needed:- For \(\cos 3\theta\): \[ \cos 3\theta = \cos^3 \theta - 3 \cos \theta (1 - \cos^2 \theta) = 4\cos^3 \theta - 3\cos \theta \]- For \(\sin 3\theta\): \[ \sin 3\theta = 3(1 - \sin^2 \theta)\sin \theta - \sin^3 \theta = 3\sin \theta - 4\sin^3 \theta \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are particularly useful in simplifying expressions and solving trigonometric equations. In the context of the exercise, we involve specific identities for \( \cos 3\theta \) and \( \sin 3\theta \). These identities help transform complex trigonometric expressions into simpler, more manageable forms.

Here we derived the identities:

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

These identities come from applying algebraic manipulations, such as the binomial theorem and double angle identities, converting complex expressions into recognizable patterns. By substituting alternative forms for \(\sin^2 \theta\) and \(\cos^2 \theta\), these trigonometric functions are streamlined. It's vital to remember these identities as they simplify the understanding of trigonometric waves and interactions.

De Moivre's Theorem

De Moivre's Theorem connects complex numbers and trigonometry. It states that for any real number \(\theta\) and integer \(n\), the expression \((\cos \theta + i \sin \theta)^n\) can be simplified to \(\cos(n\theta) + i\sin(n\theta)\). This theorem is extremely useful when you are working with powers of complex numbers represented in polar form.

In the given exercise, De Moivre's Theorem allows us to predict the trigonometric identity for powers of complex numbers. In our case, for \(n=3\), we transitioned from \((\cos \theta + i \sin \theta)^3\) to \(\cos 3\theta + i\sin 3\theta\).

De Moivre's Theorem is potent because it provides a straightforward way to handle complex number exponentiation, especially where trigonometric functions are involved. It simplifies complex calculations, making it one of the foundational blocks for understanding complex analyses in polar coordinates.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that aids in expanding expressions raised to a power. It states that \((a + b)^n\) can be expanded into a sum involving terms of the form \(a^k b^{n-k}\), multiplied by binomial coefficients. This becomes handy when dealing with expressions involving sums raised to higher powers.

For the exercise, we used the binomial theorem to expand \((\cos \theta + i \sin \theta)^3\). This expansion allows us to express the complex number as a series of simpler terms:

• \(a^3 = \cos^3\theta\)
• \(3a^2b = 3i\cos^2 \theta \sin \theta\)
• \(3ab^2 = -3\cos \theta \sin^2 \theta\)
• \(b^3 = -i\sin^3 \theta\)

Through these expansions, we can manage each component separately and ultimately derive identities for \(\cos 3\theta\) and \(\sin 3\theta\). Thus, the binomial theorem acts as a bridge that simplifies our path from complex expressions to understandable segments.

Polar Form

Polar form is a way of representing complex numbers, using a combination of lengths and angles rather than the traditional rectangular (Cartesian) coordinates. In polar form, a complex number is expressed as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude, and \(\theta\) is the angle.

In our exercise, polar form was crucial for using De Moivre's Theorem. It allowed us to translate the expression \((\cos \theta + i\sin \theta)^3\) into a trigonometric identity. This form is particularly useful because it provides a deeper insight into the geometric representation of complex numbers, making geometrical operations like rotation easier.

Understanding polar form and its use in converting complex number operations into trigonometric operations helps students draw connections between algebra and geometry, which is instrumental in advanced mathematical studies.