Problem 58 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$-5\left(\cos 320^{\circ}+i \sin 320^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(-3.830 + 3.214i\).

Step 1: Understand the Polar Form

The complex number is given in the polar form \(-5 \left( \cos 320^{\circ} + i \sin 320^{\circ} \right)\). In this form, \(-5\) is the modulus and \(320^{\circ}\) is the argument.

Step 2: Calculate Cosine and Sine

Use a calculator to find \(\cos 320^{\circ}\) and \(\sin 320^{\circ}\). Remember that \(320^{\circ}\) is in the fourth quadrant where cosine is positive and sine is negative. \(\cos 320^{\circ} = \cos (360^{\circ} - 40^{\circ}) = \cos 40^{\circ}\)\(\sin 320^{\circ} = \sin (360^{\circ} - 40^{\circ}) = -\sin 40^{\circ}\)Using a calculator, this gives approximately:- \(\cos 320^{\circ} \approx 0.7660\)- \(\sin 320^{\circ} \approx -0.6428\).

Step 3: Multiply by Modulus

Multiply both the real and the imaginary components by the modulus \(-5\):\(-5 \cdot 0.7660 = -3.830\)\(-5 \cdot (-0.6428) = 3.214\)

Key Concepts

Polar Form

In mathematics, complex numbers can often be represented in different forms. The polar form is a powerful way of expressing complex numbers through the use of a radius and an angle. It pairs the modulus with the argument for a succinct representation.
In polar form, a complex number is written as \( r (\cos \theta + i \sin \theta) \). Here, \( r \) is the modulus, representing the distance from the origin in the complex plane. The angle \( \theta \) is the argument, signifying the direction of the line from the origin.
This form is useful for multiplying and dividing complex numbers, as it simplifies these operations through straightforward arithmetic of moduli and arguments. Polar form provides a unique way to see complex numbers in terms of magnitude and direction rather than solely a mix of real and imaginary parts.

Modulus

The modulus, denoted as \( r \) in polar form, is a non-negative real number that reflects the distance of the complex number from the origin on the complex plane. For a complex number \( a + bi \), the modulus \( r \) is calculated as \( \sqrt{a^2 + b^2} \).
In the polar representation \(-5(\cos 320^{\circ} + i\sin 320^{\circ})\), the modulus is \(-5\). However, by convention, the modulus is expressed as a positive value representing magnitude, so we use the absolute value.
Think of the modulus as the radius of a circle centered at the origin. It tells us how far the complex number extends from the point (0,0). This concept is quite intuitive when you visualize complex numbers on a plane.

Argument

The argument of a complex number, marked as \( \theta \), is the angle measured from the positive x-axis to the line segment joining the origin with the point \( (a,b) \) in the complex plane.
This angle is crucial because it denotes the direction of the complex number. For the complex number \(-5(\cos 320^{\circ} + i\sin 320^{\circ})\), the argument is \(320^{\circ}\). This angle tells us where on the circle the complex number lies, using the origin as the circle's center.
Arguments are usually expressed in degrees or radians and are particularly useful in trigonometric calculations involving complex numbers. In practical problems, it's important to remember the quadrant in which the angle falls, as this influences the sign of the sine and cosine components.