Problem 51 - Precalculus

Question

Express each complex number in rectangular form. $$6\left[\cos \left(\frac{3 \pi}{4}\right)+i \sin \left(\frac{3 \pi}{4}\right)\right]$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Polar Representation

The complex number is given in the polar form \[ r \left(\cos(\theta) + i \sin(\theta)\right) \]where \( r = 6 \) and \( \theta = \frac{3\pi}{4} \). We need to convert this to rectangular form, \( a + bi \).

Step 2: Calculate the Real Part

To find the real part \( a \), use the formula:\[ a = r \cos(\theta) \]Plug in the values:\[ a = 6 \cos\left(\frac{3\pi}{4}\right) \]Since \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \),\[ a = 6 \times -\frac{\sqrt{2}}{2} = -3\sqrt{2} \]

Step 3: Calculate the Imaginary Part

For the imaginary part \( b \), use:\[ b = r \sin(\theta) \]Plug in the values:\[ b = 6 \sin\left(\frac{3\pi}{4}\right) \]Since \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \),\[ b = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} \]

Step 4: Combine Real and Imaginary Parts

Now combine the calculated real and imaginary parts to express the complex number in rectangular form:\[ -3\sqrt{2} + 3\sqrt{2}i \]

Key Concepts

Rectangular Form

The rectangular form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part of the complex number. This form is often represented on a two-dimensional plane, the complex plane, where the horizontal axis is the real axis, and the vertical axis is the imaginary axis.

It provides a clear visual representation of complex numbers and is widely used in complex arithmetic and algebra.

• Real Part (\(a\)): Represents the horizontal position on the complex plane.
• Imaginary Part (\(b\)): Represents the vertical position on the complex plane.

To convert a complex number from polar to rectangular form, use the formulas:

• \(a = r \cos(\theta)\)
• \(b = r \sin(\theta)\)

This allows for easy calculations within the complex number system, especially for addition and subtraction.

Polar Representation

Polar representation is another way to express complex numbers, emphasizing their magnitude and direction. In this form, a complex number is represented as \( r(\cos(\theta) + i \sin(\theta)) \), where \( r \) is the magnitude (or modulus), and \( \theta \) is the angle (or argument) measured from the positive real axis.

This representation is particularly useful in multiplication and division of complex numbers and is derived from the representation of a point in terms of its distance from the origin and its angle relative to the positive x-axis.

• Magnitude (\(r\)): Distance from the origin to the point, calculated as \(\sqrt{a^2 + b^2}\).
• Angle (\(\theta\)): Measured in radians, computed using \( \tan^{-1}(\frac{b}{a})\).

Understanding polar representation helps in analyzing sinusoidal signals and functions often found in engineering and physics.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit \(i\), represented as \(b\) in the standard form \(a + bi\). The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This part is crucial for calculations involving complex numbers, especially when visualizing or interpreting them in the complex plane.

Imaginary numbers extend the one-dimensional line of real numbers into the two-dimensional complex plane, giving rise to the system of complex numbers.

• Imaginary numbers can be added or subtracted, just like real numbers, i.e., \(bi + di = (b+d)i\).
• Multiplying two imaginary numbers results in a real number, as \(i \times i = -1\).

In the context of the given exercise, the imaginary part derived through the equation \[ b = r \sin(\theta) \] plays a significant role in fully understanding how the polar form translates to the rectangular form.