Problem 29 - 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 \(\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})\).

Step 1: Identify Components of Complex Number

The complex number given is \(1+i\), which can be expressed in the form \(a + bi\) where \(a = 1\) and \(b = 1\).

Step 2: Calculate Magnitude of the Complex Number

The magnitude \(r\) of the complex number \(1+i\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\). Substitute \(a = 1\) and \(b = 1\) into the formula: \[r = \sqrt{1^2 + 1^2} = \sqrt{2}\approx 1.4142\]

Step 3: Determine Angle \(\theta\)

The angle \(\theta\) for the complex number \(1+i\) is calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Substitute \(a = 1\) and \(b = 1\):\[\theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4}\]This angle is within the desired range of \(0\) to \(2\pi\).

Step 4: Write the Polar Form

The polar form of a complex number is given by \(r(\cos\theta + i\sin\theta)\). Substitute the magnitude \(r = \sqrt{2}\) and angle \(\theta = \frac{\pi}{4}\):\[\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)\]

Key Concepts

Complex Number

A complex number might sound a bit complicated at first, but it's actually quite straightforward.
Each complex number can be expressed in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part multiplied by the imaginary unit \(i\). This unit \(i\) is a special number that satisfies \(i^2 = -1\).
To visualize it: if you imagine a coordinate plane, the real part \(a\) sits on the horizontal axis, while the imaginary part \(b\) is on the vertical axis, forming a point that represents the complex number.

• Example: The complex number \(1 + i\) has a real part \(a=1\), and an imaginary part \(b=1\).
• Any complex number is a combination of a real number and an imaginary number.

Magnitude of Complex Numbers

The magnitude of a complex number is like finding the length of the diagonal in a right-angled triangle, where the real and imaginary parts are the two perpendicular sides.
It’s calculated using a formula similar to the Pythagorean theorem: \[ r = \sqrt{a^2 + b^2} \] This gives you the distance of the complex number from the origin on the complex plane.
For the number \(1+i\), plug in \(a = 1\) and \(b = 1\) into the formula to get:

• \(r = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4142\)

This magnitude tells us how "far" the complex number is from the origin.

Angle Calculation in Polar Coordinates

The angle \(\theta\) in the polar form of a complex number represents the direction of the complex number from the origin in the complex plane.
It's discovered using the formula: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] This formula is rooted in basic trigonometry concepts.The angle tells us the direction relative to the positive real axis. For instance, in the case of the complex number \(1+i\):

• You insert \(a = 1\) and \(b = 1\) into the formula to find: \( \theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \).
• This gives us an angle in radians, which is a way to express angles commonly used in mathematics.

The angle is always taken in the range of \(0\) to \(2\pi\), ensuring it's within one complete circular path around the origin.