Problem 54 - Algebra and Trigonometry

Question

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$ \left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{3} $$

Step-by-Step Solution

Answer

The result in rectangular form is \(4\sqrt{3} + 4i\).

Step 1: Apply DeMoivre's Theorem

Applying DeMoivre's Theorem to \( \left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{3}\), we get \((2)^{3}\left(\cos (3*10^{\circ})+i \sin (3*10^{\circ})\right) = 8\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\)

Step 2: Convert to Rectangular Form

The rectangular form of a complex number is given by a + bi where a is the real part and b is the imaginary part. Convert the number \(\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\) into rectangular form to obtain a = \(\cos(30^\circ)\) and b = \(\sin(30^\circ)\). The values for these Trigonometric Functions on this angle are well known, as \(\cos(30^\circ) = \sqrt{3}/2\) and \(\sin(30^\circ) = 1/2\). Substituting these values, our complex number becomes \(8(\sqrt{3}/2 + i/2) = 4\sqrt{3} + 4i\)

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The letter \(i\) is used to represent the imaginary unit, which is defined as \(i = \sqrt{-1}\).

Complex numbers are useful in various fields of mathematics and engineering as they allow us to perform calculations that include the square roots of negative numbers. They can be visualized on a complex plane, with the real part corresponding to the x-axis and the imaginary part to the y-axis. Understanding complex numbers is fundamental as they provide the building blocks for more intricate mathematical concepts like DeMoivre's Theorem.

Rectangular Form

The rectangular form of a complex number is a straightforward method to express any complex number. It is given by \(a + bi\), where \(a\) is the real component, and \(b\) is the imaginary component.

This form is particularly useful because it directly relates to the Cartesian coordinate system. By using the rectangular form, you can easily perform arithmetic operations such as addition, subtraction, multiplication, and division. For instance, the complex number \(4\sqrt{3} + 4i\) is in rectangular form, where \(4\sqrt{3}\) is the real part, and \(4\) is the coefficient of the imaginary component \(i\).

• Visualizes complex numbers on a 2D plane.
• Makes operations easily manageable.
• Directly corresponds with Cartesian coordinates.

Using the rectangular form is essential when converting results from other forms, like polar or exponential forms, into more manageable expressions.

Trigonometric Functions

Trigonometric functions such as cosine and sine are integral to working with complex numbers, especially when they are expressed in polar form. In this context, a complex number is often represented as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) the argument or angle.

These functions are fundamental because they enable the transformation between different forms of complex numbers. They are also crucial when applying DeMoivre's Theorem, as shown in the solution process. The trigonometric values for standard angles, such as 30°, 45°, and 60°, are easily obtainable:

• \(\cos(30^{\circ}) = \sqrt{3}/2\)
• \(\sin(30^{\circ}) = 1/2\)

Knowing these values allows for an easy conversion from polar to rectangular form, facilitating simpler arithmetic operations.

Powers of Complex Numbers

Calculating the powers of complex numbers becomes much simpler with DeMoivre's Theorem. This theorem states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), raising it to the \(n\)-th power involves raising \(r\) to \(n\) and multiplying the angle \(\theta\) by \(n\).

For example, when computing \(\left[2(\cos 10^{\circ}+i \sin 10^{\circ})\right]^{3}\), according to DeMoivre's Theorem:

• Power of magnitude: \(2^3 = 8\)
• Angle multiplication: \(3 \times 10^{\circ} = 30^{\circ}\)

The result \(8(\cos 30^{\circ} + i \sin 30^{\circ})\) converts this complex power calculation into manageable trigonometric and arithmetic operations. This theorem streamlines handling higher powers of complex numbers, making them accessible even when they involve intricate angles or magnitudes.