Problem 43 - Precalculus

Question

Express each complex number in rectangular form. $$2\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Trigonometric Expressions

Recognize that the expression is in polar form: \[ z = r \cdot (\cos \theta + i \sin \theta) \] Given, \( r = 2 \) and \( \theta = 315^{\circ} \).

Step 2: Calculate \( \cos 315^{\circ} \)

Understanding that \( 315^{\circ} \) corresponds to the angle \( 360^{\circ} - 45^{\circ} \), meaning it lies in the fourth quadrant where cosine is positive:\[ \cos 315^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \]

Step 3: Calculate \( \sin 315^{\circ} \)

Given that \( 315^{\circ} \) is in the fourth quadrant where sine is negative:\[ \sin 315^{\circ} = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2} \]

Step 4: Substitute Values in Rectangular Form

Substitute \( \cos 315^{\circ} \) and \( \sin 315^{\circ} \) into the polar form to express in rectangular form:\[ z = 2 \left( \frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right) \right) \]

Step 5: Simplify the Expression

Simplify the expression by distributing the \( 2 \):\[ z = \sqrt{2} - i \sqrt{2} \]

Key Concepts

Polar Form

Complex numbers can be represented in various ways, and one of these forms is the polar form. The polar form is expressed as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude of the complex number, and \( \theta \) is the angle measured in radians or degrees from the positive x-axis. This form is quite useful when dealing with complex number multiplication and division, as it can simplify these operations significantly.

• \( r \): The modulus or absolute value, represents the distance from the origin to the point.
• \( \theta \): Known as the argument, indicates the angle the line makes with the positive real axis.

Understanding polar form can provide deeper insights into the geometric interpretation of complex numbers, showing them as vectors with a direction and magnitude.

Rectangular Form

While polar form is great for certain mathematical operations, the rectangular form is often more intuitive and easier to use in calculation.In rectangular form, a complex number is written as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The transition from polar to rectangular form, seen in the exercise, involves evaluating trigonometric expressions and involves basic algebraic operations.

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

These formulas convert the polar coordinates into rectangular coordinates, making it simple to express complex numbers in a way that is often encountered in algebra and calculus problems.

Trigonometric Expressions

Trigonometry plays a crucial role in converting between polar and rectangular forms. In the exercise, you encounter \( \cos 315^{\circ} \) and \( \sin 315^{\circ} \), which are fundamental trigonometric expressions.These expressions are used to find the values of \( x \) and \( y \) coordinates for a given angle.

• \( \cos 315^{\circ} = \frac{\sqrt{2}}{2} \), since this angle lies in the fourth quadrant where cosine is positive.
• \( \sin 315^{\circ} = -\frac{\sqrt{2}}{2} \), reflecting the fact that sine is negative in the fourth quadrant.

These trigonometric calculations are essential for accurately converting between different complex number forms. Understanding how to manipulate these expressions enables solving a wide range of mathematical problems involving angles and distances in the complex plane.