Problem 126 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[4(\cos 2.8+i \sin 2.8)]^{5}$$

Step-by-Step Solution

Answer

The result of raising the given complex number to the power 5 is \( 1024[cos(14) + i sin(14)] \).

Step 1: Identify the Variables in The Complex Number

The complex number is in the form \( 4(cos 2.8 + i sin 2.8) \). Here, \( r = 4 \), \( \theta = 2.8 \), and \( n = 5 \).

Step 2: Use DeMoivre's Theorem

Apply DeMoivre's Theorem to raise the complex number to the power \( 5 \). According to DeMoivre's theorem, \( [r(cos \theta + i \sin \theta)]^n = r^n ( cos(n\theta) + i \sin(n\theta))\). So, this gives us \( [4(cos 2.8 + i \sin 2.8)]^5 = 4^5[cos(5*2.8) + i sin(5*2.8)] \)

Step 3: Simplify Results

Now simplify the expression: \( 4^5 = 1024 \) and \( 5*2.8 = 14 \). Substitute these values back to get : \( 1024[cos(14) + i sin(14)] \)

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical constructs, consisting of two main parts — a real part and an imaginary part. These numbers extend our regular number system by including the imaginary unit, denoted as \(i\), which is defined by the equation \(i^2 = -1\). By incorporating \(i\), complex numbers help solve equations that do not have real solutions. For instance, the equation \(x^2 + 1 = 0\) can be solved using complex numbers because \(x\) can equal \(i\) or \(-i\).

In general, a complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real component, and \(b\) times \(i\) represent the imaginary component. Together, they form \(z = a + bi\), a complex number. These numbers are incredibly useful in various fields of science and engineering, particularly when dealing with oscillations and waves.

Polar Form

To make calculations easier, particularly when finding powers or roots of complex numbers, it can be beneficial to express them in polar form. The polar form of a complex number converts it into a format that combines a magnitude and an angle, essentially representing the number on a 2D plane.

A complex number \(z = a + bi\) can be rewritten in polar form as \(r(\cos \theta + i \sin \theta)\), where:

• \(r\) is the magnitude or modulus, calculated as \(\sqrt{a^2 + b^2}\).
• \(\theta\) is the argument, which represents the angle formed with the positive x-axis and is computed using \(\tan^{-1}(b/a)\).

Polar form is particularly helpful because when multiplying or raising to powers, you only need to focus on the angles and magnitudes, simplifying the process significantly.

Radian Measure

In mathematics, especially trigonometry, angles can be expressed in degrees or radians. Radian measure is a more natural unit for many mathematical computations because it directly relates the angle to the arc length and radius of a circle.

A radian is based on the radius of a circle. When an angle's subtended arc length equals the radius, the angle measures exactly 1 radian. This makes it different than degrees, where a full circle equals 360°. In radians, a full circle is \(2\pi\) radians. Therefore, to convert degrees to radians, you can use the formula: \(\text{radians} = \frac{\pi}{180} \times \text{degrees}\).

Radian measure is widely used in calculus, physics, and engineering due to its convenience in applying trigonometric identities and formulas.

Powers of Complex Numbers

Raising complex numbers to a power can initially seem complicated, but DeMoivre's Theorem makes these calculations manageable. When a complex number is expressed in polar form, raising it to a power simply involves using DeMoivre's Theorem. This theorem states:

• \([r(\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta))\)

Here:

• \(r^n\) is the magnitude raised to the power \(n\).
• \(n\theta\) is the angle multiplied by \(n\).

After calculating, the result can be converted back into its standard complex form \(a + bi\), if needed. This process provides a straightforward way to work with the powers of complex numbers, streamlining calculations that would otherwise be cumbersome.