Problem 48 - Precalculus

Question

Express each complex number in rectangular form. $$\sqrt{3}\left(\cos 330^{\circ}+i \sin 330^{\circ}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify Given Expression

The expression given is presented in polar form: \( z = r (\cos \theta + i \sin \theta) \), where \( r = \sqrt{3} \) and \( \theta = 330^{\circ} \).

Step 2: Recall Rectangular Form

The rectangular form of a complex number is \( x + yi \), where \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Calculate \( \cos 330^{\circ} \) and \( \sin 330^{\circ} \)

\( \cos 330^{\circ} \) and \( \sin 330^{\circ} \) correspond to angles on the unit circle. We know that \( \cos 330^{\circ} = \cos (-30^{\circ}) = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 330^{\circ} = \sin (-30^{\circ}) = -\sin 30^{\circ} = -\frac{1}{2} \).

Step 4: Substitute Values and Compute \( x \) and \( y \)

Substitute \( \cos 330^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 330^{\circ} = -\frac{1}{2} \) into the formulas: \[ x = \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3}{2} \]\[ y = \sqrt{3} \cdot -\frac{1}{2} = -\frac{\sqrt{3}}{2} \]

Step 5: Write the Rectangular Form

Combine the computed values to express the complex number in rectangular form. Thus, the rectangular form is: \[ \frac{3}{2} - \frac{\sqrt{3}}{2}i \]

Key Concepts

Rectangular Form

Complex numbers can be expressed in various forms, each serving different purposes. One of the simplest to understand is the rectangular form, which represents complex numbers as the sum of a real part and an imaginary part. This form is given by the equation:

• \( z = x + yi \)
• where \( x \) represents the real part and \( y \) represents the imaginary part.

To convert a complex number from polar to rectangular form, we identify \( x \) and \( y \) using trigonometric identities:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This means that on the coordinate plane, \( x \) is the horizontal distance from the origin, while \( y \) is the vertical distance. By converting to rectangular form, we see directly where a complex number "lives" on the complex plane.

Polar Form

When dealing with complex numbers, another common representation is the polar form. This is often useful in mathematical calculations involving rotation and phase shifts. A complex number in polar form is denoted as:

• \( z = r ( \cos \theta + i \sin \theta ) \)
• where \( r \) represents the magnitude or modulus of the complex number, and \( \theta \) is the angle or argument.

The magnitude \( r \) is the "distance" from the origin to the point on the plane, and can be found using the Pythagorean theorem. The angle \( \theta \), often in degrees or radians, specifies the direction.
This form is particularly handy for multiplying complex numbers, as it allows us to multiply magnitudes and add angles directly.

Trigonometric Functions

Understanding trigonometric functions is key to converting between polar and rectangular forms and vice versa. In the context of complex numbers:

• \( \cos \theta \) (cosine) represents the adjacent side of a right triangle divided by the hypotenuse. It gives the horizontal component relative to the angle \( \theta \).
• \( \sin \theta \) (sine) represents the opposite side divided by the hypotenuse, providing the vertical component.

In the problem, the angle 330° was used because it provides a natural extension to negative angles in standard position. We calculate cosine and sine of this angle to find the rectangular components. These functions are cyclical and help navigate the unit circle to determine coordinates that translate into \( x \) and \( y \) in the real and imaginary parts of the complex number. Mastery of these concepts allows for seamless transitions between different forms of complex numbers.