Problem 45 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 3 i(1+i) $$

Step-by-Step Solution

Answer

In polar form: \(3\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\).

Step 1: Multiply the Complex Numbers

First, we need to multiply the two complex numbers, which are \(3i\) and \((1+i)\). Recall that \(i^2 = -1\). So, \[3i \times (1 + i) = 3i \times 1 + 3i \times i\]\[= 3i + 3i^2\]\[= 3i + 3(-1)\]\[= 3i - 3\]Thus, the resulting complex number is \(-3 + 3i\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by the formula \[r = \sqrt{a^2 + b^2}\]For our complex number, \(a = -3\) and \(b = 3\), so \[r = \sqrt{(-3)^2 + 3^2} \]\[= \sqrt{9 + 9}\]\[= \sqrt{18}\]\[= 3\sqrt{2}\].

Step 3: Determine the Argument

The argument \(\theta\) of a complex number \(a + bi\) is determined by the arctangent of the quotient of imaginary and real parts:\[\theta = \arctan\left(\frac{b}{a}\right)\]For \(-3 + 3i\), we have \[\theta = \arctan\left(\frac{3}{-3}\right) = \arctan(-1)\]Given \(-3\) is negative and \(3\) is positive, the complex number lies in the second quadrant where \(\theta = \pi - \arctan(1)\):\[\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\].

Step 4: Represent in Polar Form

Finally, we write the polar form as \[r(\cos \theta + i \sin \theta)\]Substituting the values for \(r\) and \(\theta\), we have \[3\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\]

Key Concepts

Polar Form

The polar form of a complex number can resemble a very different perspective compared to the usual rectangular form. Rather than focusing on the real part and the imaginary part separately, polar form emphasizes the magnitude and direction. This is particularly useful in problems involving complex multiplication or division.

In polar coordinates, any complex number is written as \[ r(\cos \theta + i \sin \theta) \] where:

• \( r \) is the magnitude of the complex number, determining its distance from the origin in the complex plane
• \( \theta \) is the argument, representing the angle formed with the positive real axis

Expressing a number in polar form is about translating from a grid-like, rectangular map (real and imaginary parts) to a circular, compass-like map (magnitude and angle). This view gives us insight into how complex numbers behave under multiplication and division, using the concepts of rotation and scaling.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as its absolute value or modulus, is a measure of its size or length, neglecting its direction. Imagine plotting the complex number \( a + bi \) on a plane. The magnitude is the distance from the origin to the point \( (a, b) \).

Mathematically, the magnitude \( r \) is calculated using the Pythagorean theorem:

• \( r = \sqrt{a^2 + b^2} \)

This is equivalent to computing the hypotenuse of a right triangle where the legs are aligned with the real and imaginary components of the number.

For instance, the complex number \(-3 + 3i\) has a magnitude:

• \( r = \sqrt{(-3)^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \)

Magnitude tells us how 'far' a complex number is from the origin, giving us a sense of its "strength" without regard to its orientation in the complex plane.

Argument of Complex Numbers

The argument of a complex number is like its compass direction. It shows which way we need to "look" from the origin to locate the number in the complex plane. The argument is expressed in radians, taking values between \(0\) and \(2\pi\).

To find the argument \( \theta \) of a complex number \( a + bi \) , we use the arctangent of the ratio of the imaginary to the real part:

• \( \theta = \arctan\left(\frac{b}{a}\right) \)

However, special care must be taken regarding the sign of \( a \) and \( b \), as this determines the quadrant.

For \(-3 + 3i\), since the real part is negative and the imaginary part is positive, the number is located in the second quadrant. In this case:

• \( \theta = \pi - \arctan(1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)

This understanding of the argument provides a full picture of the complex number's direction relative to the positive real axis, completing its polar representation.