Problem 32 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$-3+4 i$$

Step-by-Step Solution

Answer

The polar form is \(5\left(\cos(\pi - \tan^{-1}(\frac{4}{3}))+i\sin(\pi - \tan^{-1}(\frac{4}{3}))\right)\).

Step 1: Identify the Real and Imaginary Parts

The complex number given is \(-3 + 4i\).Identify the real part as \(-3\) and the imaginary part as \(4\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \( a + bi \) is calculated using the formula \(r = \sqrt{a^2 + b^2} \).For \(-3 + 4i\), compute \(r = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Step 3: Find the Argument (Angle)

The argument \( \theta \) is found using \( \tan(\theta) = \frac{b}{a} \).Substitute the values \( a = -3 \) and \( b = 4 \) to get \( \tan(\theta) = \frac{4}{-3} \), which implies that \( \theta = \tan^{-1}\left(-\frac{4}{3}\right)\).Since the complex number is in the second quadrant (negative real, positive imaginary), adjust the angle to \( \theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)\).

Step 4: Convert to Polar Form

The polar form of a complex number is \(r(\cos\theta + i\sin\theta)\), or using Euler's formula, \(re^{i\theta}\).Substitute the magnitude \( r = 5 \) and the angle \( \theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)\) to obtain the polar form.Thus, \(-3 + 4i = 5\left(\cos(\pi - \tan^{-1}\left(\frac{4}{3}\right)) + i\sin(\pi - \tan^{-1}\left(\frac{4}{3}\right))\right)\).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part with \(i\) representing the square root of -1. This special number \(i\) is what makes complex numbers unique since it allows us to expand the number system to include solutions to equations that don't have real solutions. For instance, in the complex number \(-3 + 4i\), the real part is \(-3\), and the imaginary part is \(4i\). This form is known as the rectangular form of a complex number because it can be visually represented on a two-dimensional plane, known as the complex plane. This plane has a horizontal axis for the real part and a vertical axis for the imaginary part.

Magnitude of Complex Numbers

The magnitude of a complex number, sometimes called the modulus, is a measure of its size. If you think of complex numbers as vectors on the complex plane, the magnitude represents the distance from the origin \((0,0)\) to the point \((a, b)\).To find this magnitude, we use the Pythagorean Theorem, which is expressed through the formula: - \(r = \sqrt{a^2 + b^2}\)This formula calculates the distance by treating the real part \(a\) and the imaginary part \(b\) as the x and y components of a right-angled triangle. For example, the magnitude of \(-3 + 4i\) is calculated as:- \(r = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)Having the magnitude allows us to express complex numbers in polar form, a different way to view them with a focus on this distance and angle.

Argument of Complex Numbers

The argument of a complex number gives us the angle direction of the number from the positive real axis, measured in a counter-clockwise direction. It is denoted by \(\theta\) and can be found using trigonometric functions like tangent.Given that \( \tan(\theta) = \frac{b}{a} \), you can calculate the angle for the complex number. However, due to the cyclical nature of the tangent function and the location of the complex number on the plane, sometimes adjustments are necessary.For \(-3 + 4i\), since it resides in the second quadrant (where the real part is negative and the imaginary part is positive), the initial calculation using \( \tan(\theta) = \frac{4}{-3} \) gives us \( \theta = \tan^{-1}\left(-\frac{4}{3}\right)\). Yet, to locate it in the second quadrant accurately, we adjust by using:- \(\theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)\)This corrects the angle to its proper position on the complex plane.

Euler's Formula

Euler's Formula is an elegant way to express complex numbers in polar form. It connects the rectangular coordinates with trigonometric functions and exponential functions, showing the deep relationship between these branches of mathematics.Euler's Formula states that for a complex number, - \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\)Thus, a complex number in polar form \( r(\cos\theta + i\sin\theta)\) can also be written using Euler's Formula as \( re^{i\theta} \).Applying this to \(-3 + 4i\), once we know the magnitude \( r = 5 \) and the argument \(\theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)\), its polar form is:- \(5e^{i(\pi - \tan^{-1}\left(\frac{4}{3}\right))}\)This encapsulates both the distance from the origin and the direction in a compact and beautiful expression.