Question

\([\\{1+\cos (\pi / 12)+i \sin (\pi / 12)\\} /\\{1+\cos (\pi / 12)\) \(-i \sin (\pi / 12)\\}]^{36}=\) (a) \(-1\) (b) 1 (c) 0 (d) \((1 / 2)\)

Short answer

The given complex expression simplifies to \(1\), and when raised to the power of \(36\), it remains \(1\), which corresponds to option (b).

Step-by-step solution

Simplify the Expression

To start, let's simplify the given expression: \[ \frac{1+\cos(\frac{\pi}{12})+i\sin(\frac{\pi}{12})}{1+\cos(\frac{\pi}{12})-i\sin(\frac{\pi}{12})} \] One way to rationalize this complex number is to multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{(1+\cos(\frac{\pi}{12})+i\sin(\frac{\pi}{12})) (1+\cos(\frac{\pi}{12})+i\sin(\frac{\pi}{12}))}{(1+\cos(\frac{\pi}{12})-i\sin(\frac{\pi}{12}))(1+\cos(\frac{\pi}{12})+i\sin(\frac{\pi}{12}))} \]

Multiply and Simplify

Next, we multiply and simplify the expression: \[ \frac{1+2\cos(\frac{\pi}{12})+\cos^2(\frac{\pi}{12}) - \sin^2(\frac{\pi}{12})}{1+2\cos(\frac{\pi}{12})+\cos^2(\frac{\pi}{12})+\sin^2(\frac{\pi}{12})} \] Now, note that \(\cos^2(\frac{\pi}{12})+\sin^2(\frac{\pi}{12})=1\) . So, we can replace this sum with 1: \[ \frac{1+2\cos(\frac{\pi}{12})+(1- \sin^2(\frac{\pi}{12}))}{1+2\cos(\frac{\pi}{12})+1} \] Further simplifying, we obtain: \[ \frac{2(1+\cos(\frac{\pi}{12}))}{2+2\cos(\frac{\pi}{12})} \]

Cancel terms

Now, cancel the common terms in the numerator and the denominator: \[ \frac{1+\cos(\frac{\pi}{12})}{1+\cos(\frac{\pi}{12})} = 1 \]

Raise to the 36th power

Now, we need to raise the result we found in the previous step to the power of 36: \[ (1)^{36} = 1 \] The result matches option (b) which is 1.

Key concepts

Complex Conjugate

A complex conjugate is a pair of complex numbers with the same real part and opposite imaginary parts. So if you have a complex number written as \(a + bi\), its complex conjugate would be \(a - bi\).

The conjugate plays an important role when we work with complex numbers. It helps in simplifying division of complex numbers, which is exactly what we see in step one of the solution. By multiplying the numerator and the denominator by the conjugate of the denominator, you turn the denominator into a real number.

Let's simplify: If our denominator is \(1 + \cos(\frac{\pi}{12}) - i\sin(\frac{\pi}{12})\), the conjugate will be \( 1 + \cos(\frac{\pi}{12}) + i\sin(\frac{\pi}{12}) \). This makes calculation easier because multiplying these results in the sum of squares of the trigonometric functions — a technique often used in solving complex fraction problems efficiently.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool used in complex number calculations, especially when raising them to powers and finding roots. It's particularly useful for expressions involving trigonometric functions. Specifically, it states that \[(r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\] where \(r\) is the modulus of the complex number and \(\theta\) is the argument.

In our problem, however, since we simplified the complex expression to 1, we didn't need to apply De Moivre's directly for the exponential part. But if we had a complex number with real and imaginary components, De Moivre's Theorem would allow us to easily compute the power of that complex number. This demonstrates the utility of the theorem in dealing with more complex issues when they arise in problems like the one we've seen.

Trigonometric Functions

Trigonometric functions, namely sine, cosine, and tangent, play crucial roles in both real and complex analysis. In complex numbers, they help express complex exponentials and are key components when simplifying expressions.

In the context of this exercise, trigonometric identities like \(\cos^2(\theta) + \sin^2(\theta) = 1\) are applied to simplify expressions that arise when working with complex numbers. For example, in step two of our solution, this identity was crucial in reducing a more complicated expression to a simpler form.

These trigonometric rules and identities help bridge the gap between algebraic and geometric interpretations of problems, enabling solutions to be found more methodically and with less computational work.