Problem 47 - Precalculus

Question

Express each complex number in rectangular form. $$\sqrt{3}\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is \(-\frac{3}{2} + \frac{\sqrt{3}}{2}i\).

Step 1: Recognize polar form

The given complex number is in polar form: \(r (\cos \theta + i \sin \theta)\). Here, \(r = \sqrt{3}\) and \(\theta = 150^{\circ}\). We need to convert this into rectangular form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Calculate real part

The real part \(a\) is given by \(a = r \cos \theta\). Substitute \(r = \sqrt{3}\) and \(\theta = 150^{\circ}\): \[ a = \sqrt{3} \cos 150^{\circ} = \sqrt{3} \times \left( -\frac{\sqrt{3}}{2} \right) = -\frac{3}{2} \]

Step 3: Calculate imaginary part

The imaginary part \(b\) is given by \(b = r \sin \theta\). Substitute \(r = \sqrt{3}\) and \(\theta = 150^{\circ}\): \[ b = \sqrt{3} \sin 150^{\circ} = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \]

Step 4: Combine into rectangular form

Now that we have both the real and imaginary parts, we can express the complex number in rectangular form: \[ -\frac{3}{2} + \frac{\sqrt{3}}{2}i \]

Key Concepts

Rectangular Form

Rectangular form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form allows us to easily identify the components of a complex number by simply observing the equation. It is called "rectangular" because it can be represented on a two-dimensional plane, with the real part as the horizontal component and the imaginary part as the vertical component.

• Real Part \(a\): This is the horizontal distance from the origin to the point on the complex plane, analogous to the x-coordinate in Cartesian coordinates.
• Imaginary Part \(b\): This is the vertical distance, similar to the y-coordinate.

Understanding the rectangular form is crucial for performing operations such as addition and subtraction on complex numbers, as it involves straightforward arithmetic operations on the separate real and imaginary parts. This form simplifies visualizing complex numbers as points or vectors in the plane.

Polar Form

Polar form of a complex number expresses it in terms of a magnitude and an angle, indicated as \(r(\cos \theta + i \sin \theta)\) or \(re^{i\theta}\). Here, \(r\) is the magnitude or modulus and \(\theta\) is the angle or argument. This form is particularly useful for multiplying and dividing complex numbers, as well as finding powers and roots.

• Magnitude \(r\): Represents the distance of the point from the origin in the complex plane, calculated as \(\sqrt{a^2 + b^2}\).
• Angle \(\theta\): Also known as the argument, it denotes the direction or angle made with the positive x-axis, often measured in degrees or radians.

Polar form is especially advantageous in fields like engineering and physics when dealing with sinusoidal functions or rotating objects because it reflects both the size and direction conveniently. Additionally, it simplifies the computation of powers of complex numbers using De Moivre's theorem.

Conversion Between Forms

Converting between rectangular and polar forms is an essential skill in handling complex numbers. From polar to rectangular, we translate the expression \(r(\cos \theta + i \sin \theta)\) into \(a + bi\) by calculating the real and imaginary parts separately.

From Polar to Rectangular

To go from polar to rectangular:

• Real Part \(a\): Calculate using \(a = r \cos \theta\).
• Imaginary Part \(b\): Calculate using \(b = r \sin \theta\).

Combine these to form \(a + bi\). For example, given \(r = \sqrt{3}\) and \(\theta = 150^{\circ}\):

• \(a = \sqrt{3} \times \cos(150^{\circ}) = -\frac{3}{2}\)
• \(b = \sqrt{3} \times \sin(150^{\circ}) = \frac{\sqrt{3}}{2}\)

This gives the rectangular form: \(-\frac{3}{2} + \frac{\sqrt{3}}{2}i\)

From Rectangular to Polar

To convert from rectangular to polar form:

• Magnitude \(r\): Compute using \(r = \sqrt{a^2 + b^2}\).
• Angle \(\theta\): Find \(\theta\) using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

Both methods allow a seamless transition between forms, facilitating various operations and simplifying complex math tasks. Recognizing when to use each form is key to unlocking the full potential of complex numbers.