Problem 34 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$-1+i$$

Step-by-Step Solution

Answer

The polar form is \\\( -1+i = \sqrt{2}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})\\).

Step 1: Identify the real and imaginary parts

The real part of the complex number is \(a = -1\) and the imaginary part is \(b = 1\). The number is written as \(-1+i\).

Step 2: Calculate the modulus

The modulus \(r\) of the complex number \(-1+i\) is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). Substitute the values: \[ r = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}. \] Hence, the modulus \(r = \sqrt{2}\).

Step 3: Determine the argument

The argument \(\theta\) is calculated using \( an^{-1}(\frac{b}{a})\). Here \(b=1\) and \(a=-1\). This gives \[ \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1). \] Since the complex number is in the second quadrant, \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Step 4: Write the polar form

Combine the modulus and argument to express the complex number in polar form. The polar form is given by \(z = r(\cos\theta + i\sin\theta)\). Thus, \[ -1+i = \sqrt{2}\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right). \]

Key Concepts

Modulus of a Complex Number

In mathematics, a complex number is typically represented as \( z = a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part. To find the modulus of a complex number, we determine its distance from the origin in the complex plane. This distance, represented by \( |z| \), is calculated using the formula: \[ |z| = \sqrt{a^2 + b^2}. \] For our specific example of \( -1+i \), the real part \( a = -1 \) and the imaginary part \( b = 1 \). By substituting these values into the formula, we find: \[ |z| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}. \] Thus, the modulus of the complex number \( -1+i \) is \( \sqrt{2} \). This tells us how far \( -1+i \) is from the origin on the complex plane.

Argument of a Complex Number

The argument of a complex number gives us the angle the line representing the complex number makes with the positive real axis, when plotted on the complex plane. This angle is denoted by \( \theta \). For our example \( -1+i \), we calculate the argument using the formula: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right). \] Here, since \( a = -1 \) and \( b = 1 \), we have: \[ \theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1). \] To refine this, we must consider the quadrant where our complex number lies. In this case, \( -1+i \) is in the second quadrant (top-left portion). This alters the argument calculation. Therefore, we adjust using the formula: \[ \theta = \pi - \tan^{-1}(1) = \frac{3\pi}{4}. \] This provides the correct argument, which is important for accurately expressing the complex number in polar form.

Tangent Inverse

The tangent inverse (also denoted as \( \tan^{-1} \) or \( \arctan \)) is a trigonometric function used to determine the angle whose tangent is a given number. In our context, it helps to find the angle that a complex number's vector makes with the positive x-axis. For instance, in our calculation for the argument of \( -1+i \), the formula \( \tan^{-1}\left(\frac{b}{a}\right) \) was used. With \( b=1 \) and \( a=-1 \), the calculation yields: \[ \theta = \tan^{-1}(-1). \] The typical range of values for \( \tan^{-1} \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which must be adjusted based on the quadrant of the complex number. By understanding the function's output and the number's quadrant, we ensure the right angle is identified.

Complex Numbers in Second Quadrant

In the complex plane, numbers can be located in different quadrants based on the signs of their real and imaginary parts. The second quadrant, located in the upper-left section, is defined by negative real parts and positive imaginary parts. Here, the visual representation of \( -1+i \) shows it lies in the second quadrant. This location is important as it influences our calculation of the argument \( \theta \). For numbers in the second quadrant, \( \theta \) is calculated by subtracting the result of \( \tan^{-1} \) from \( \pi \). This adjustment accounts for the angle's directional turn over the axis: \[ \theta = \pi - \tan^{-1}(1) = \frac{3\pi}{4}. \] Understanding the quadrant helps properly express complex numbers in polar form, enabling further mathematical operations like multiplication or division of complex numbers.