Problem 8 - Calculus

Question

Find \(x\) and \(y\) for the complex numbers \(x+i y\) at angles \(\theta=45^{\circ}, 90^{\circ}, 135^{\circ}\) on the unit circle. Verify directly that the square of the first is the second and the cube of the first is the third.

Step-by-Step Solution

Answer

The complex numbers are \(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\), \(i\), and \(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\).

Step 1: Identify coordinates on the unit circle

For a unit circle, a point at an angle \( \theta \) can be represented as a complex number \( x + i y \) where \( x = \cos(\theta) \) and \( y = \sin(\theta) \). We will calculate this for the given angles: - \( \theta = 45^{\circ} \): - \( x = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), - \( y = \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \). - \( \theta = 90^{\circ} \): - \( x = \cos(90^{\circ}) = 0 \), - \( y = \sin(90^{\circ}) = 1 \). - \( \theta = 135^{\circ} \): - \( x = \cos(135^{\circ}) = -\frac{\sqrt{2}}{2} \), - \( y = \sin(135^{\circ}) = \frac{\sqrt{2}}{2} \).

Step 2: Verify the square relationship

We need to verify that the square of the first complex number is equal to the second. Square the first: \[ (\cos(45^{\circ}) + i \sin(45^{\circ}))^2 = \left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right)^2 \]Expanding the square: \[ (\frac{\sqrt{2}}{2})^2 + 2(\frac{\sqrt{2}}{2})(\frac{\sqrt{2}}{2})i + (i \frac{\sqrt{2}}{2})^2 \]This simplifies to: \[ \frac{1}{2} + i + \frac{1}{2}(-1) = i \]Which is equivalent to \(0 + i\) or the second point \(\cos(90^{\circ}) + i \sin(90^{\circ})\).

Step 3: Verify the cube relationship

We need to verify that the cube of the first complex number equals the third. Cube the first:\[ (\cos(45^{\circ}) + i \sin(45^{\circ}))^3 = \left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right)^3 \]Using De Moivre's Theorem which states \( (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \), substitute \( n = 3 \): \[ = \cos(135^{\circ}) + i \sin(135^{\circ}) \]This results in the third point \(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\). The expression matches \(\cos(135^{\circ}) + i \sin(135^{\circ})\).

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry and complex numbers. It’s a circle with a radius of 1 centered at the origin (0,0) on the coordinate plane.

In the context of complex numbers, any point on the unit circle can be represented in the form of a complex number, that is, in the form of \(x + iy\). Here, \(x\) is the real part and \(y\) is the imaginary part of the complex number.

For any angle \(\theta\), the point on the unit circle related to this angle can be given by coordinates:

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

This means that the angle \(\theta\) in radians or degrees determines the position on the unit circle. Therefore, each angle corresponds to a unique point (complex number) on the circle.

De Moivre's Theorem

De Moivre's Theorem provides a useful formula for raising complex numbers in polar form to any power. It states: \( (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \).

This theorem simplifies the process of multiplying a complex number by itself multiple times.

• For squaring, set \(n = 2\).
• For cubing, set \(n = 3\).

When using De Moivre's Theorem, ensure you're working with angles in the same unit, either degrees or radians. For instance, for an initial angle \(\theta = 45^{\circ}\):

• The square is at \(90^{\circ}\).
• The cube is at \(135^{\circ}\).

This demonstrates how efficiently De Moivre's theorem deals with powers of complex numbers versus traditional expansion methods.

Trigonometric Functions

Trigonometric functions such as cosine and sine are key to understanding the behavior of complex numbers on the unit circle. In essence, these functions project the radius vector of the unit circle onto the x-axis and y-axis respectively.

The cosine of an angle \(\theta\) tells you how far along the x-axis the point is. In contrast, the sine provides the distance along the y-axis. For instance, at \(45^{\circ}\), both \(\cos(45^{\circ})\) and \(\sin(45^{\circ})\) are \(\frac{\sqrt{2}}{2}\).

• \(\cos(90^{\circ})\) equals 0.
• \(\sin(90^{\circ})\) equals 1.
• \(\cos(135^{\circ})\) equals \(-\frac{\sqrt{2}}{2}\).
• \(\sin(135^{\circ})\) equals \(\frac{\sqrt{2}}{2}\).

These functions are periodic, meaning they repeat their values in a predictable pattern, linked to angles on the unit circle.

Angle Measurement

Angles can be measured in two main units: degrees and radians. It’s important to understand each and how they relate.

A full circle is 360 degrees, which equates to \(2\pi\) radians. Therefore, converting between degrees and radians is crucial in calculations: \(1\;\text{degree} = \frac{\pi}{180}\;\text{radian}\) and \(1\;\text{radian} = \frac{180}{\pi}\;\text{degrees}\).

When working with trigonometric functions or De Moivre's Theorem, the choice of unit can alter results if not used consistently:

• 45 degrees equals \(\frac{\pi}{4}\) radians.
• 90 degrees is \(\frac{\pi}{2}\) radians.
• 135 degrees is \(\frac{3\pi}{4}\) radians.

Understanding these conversions ensures that mathematical operations involving complex numbers are performed accurately.