Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (2+5 i)^{3} $$

Step-by-Step Solution

Answer

The expression is in the form \(a + bi\).

Step 1: Express the Complex Number in Polar Form

The complex number given is \(2 + 5i\). First, we calculate the modulus \(r\) using the formula \(r = \sqrt{a^2 + b^2}\): \[ r = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}. \] Next, find the argument \(\theta\) using \(\tan^{-1}\left(\frac{b}{a}\right)\): \[ \theta = \tan^{-1}\left(\frac{5}{2}\right). \] Now, in polar form, \(2 + 5i\) is represented as \( \sqrt{29}(\cos\theta + i\sin\theta) \).

Step 2: Use De Moivre's Theorem

According to De Moivre's Theorem, for a complex number \( r (\cos \theta + i \sin \theta) \), its \(n\)-th power is given by: \[ r^n (\cos(n\theta) + i\sin(n\theta)). \]For \((2 + 5i)^3\), substitute \(r = \sqrt{29}\) and \(n = 3\): \[ (\sqrt{29})^3 (\cos(3\theta) + i\sin(3\theta)) = 29\sqrt{29} (\cos(3\theta) + i\sin(3\theta)). \]

Step 3: Calculate the Cosine and Sine Values

To find \(3\theta\), calculate \(\theta\) and then multiply by 3. The argument \(\theta = \tan^{-1}\left(\frac{5}{2}\right)\) was computed earlier. Use this to determine \(\cos(3\theta)\) and \(\sin(3\theta)\) using a calculator or trigonometric tables.

Step 4: Convert Back to Rectangular Form

Multiply the results from Step 3 by \(29\sqrt{29}\) to obtain the rectangular form:\[a = 29\sqrt{29} \cos(3\theta) \] \[b = 29\sqrt{29} \sin(3\theta) \]Fill in the values of \(\cos(3\theta)\) and \(\sin(3\theta)\) to compute \(a\) and \(b\). Thus, the expression is in the desired form \(a + bi\).

Key Concepts

Polar Form

Complex numbers can be expressed in the form of \(a + bi\), known as the rectangular form, where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. However, there's another way known as the polar form, which can often simplify the process of multiplication and exponentiation of complex numbers.

The polar form is based on two components:

• Modulus \(r\): The distance from the origin to the point \((a, b)\) in the complex plane. It's calculated as \(r = \sqrt{a^2 + b^2}\).
• Argument \(\theta\): The angle formed with the positive x-axis, found using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

For a complex number \(2 + 5i\), its modulus \(r\) is \(\sqrt{29}\), and \(\theta\) is \(\tan^{-1}\left(\frac{5}{2}\right)\).

This transforms \(2 + 5i\) into polar form: \(\sqrt{29}(\cos\theta + i\sin\theta)\). Hence, polar form provides a bridge between geometry and algebra, offering insights into the number's magnitude and direction.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number operations, particularly when computing powers and roots. It simplifies calculations by expressing complex numbers in polar form.

• Theorem Statement: For a complex number \(r (\cos \theta + i \sin \theta)\), its \(n\)-th power is \(r^n (\cos(n\theta) + i\sin(n\theta))\).

In our problem, we raised \((2 + 5i)\) to the third power. After transforming \(2 + 5i\) into polar form \(\sqrt{29}(\cos\theta + i\sin\theta)\), De Moivre's Theorem allows us to efficiently compute \((2 + 5i)^3\) as:

\(29\sqrt{29} (\cos(3\theta) + i\sin(3\theta))\).

This theorem not only eases the process of finding higher powers of complex numbers but also retains the geometric intuition conveyed by the polar form.

Rectangular Form

To retrieve results back into the rectangular form \(a + bi\), we calculate the real and imaginary parts separately.

After finding \(n\theta\) and using a calculator or trigonometric tables, the cosine and sine values are determined. These are found by replacing the calculated angles into the following:

• \(a = r^n \cos(n\theta)\)
• \(b = r^n \sin(n\theta)\)

For \((2 + 5i)^3\), \(a = 29\sqrt{29} \cos(3\theta)\) and \(b = 29\sqrt{29} \sin(3\theta)\) which upon calculation yield real number values for \(a\) and \(b\).

This process of conversion highlights the versatility of complex numbers, allowing them to be shifted efficiently between geometric and numeric contexts for practical problem-solving.