Problem 1 - A Book of Abstract Algebra

Question

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). For each \(x \in \mathbb{R}\), it is conventional to write cis \(x=\cos x+i \sin x\). Prove that cis \((x+y)=(\operatorname{cis} x)(\) cis \(y)\).

Step-by-Step Solution

Answer

cis (x+y) = (cis x)(cis y) is proven by using trigonometric angle addition identities to show both sides are equal.

Step 1: Understand the Given Expression

We start with the expression we need to prove: \(\operatorname{cis}(x+y) = (\operatorname{cis} \, x)(\operatorname{cis} \, y)\). We know that \(\operatorname{cis} \, x = \cos x + i \sin x\) and \(\operatorname{cis} \, y = \cos y + i \sin y\).

Step 2: Substituting into the Left Hand Side

Substitute \(x+y\) into \(\operatorname{cis}(x+y)\):\[ \operatorname{cis}(x+y) = \cos(x+y) + i\sin(x+y) \]

Step 3: Expand the Trigonometric Functions

Use the angle sum identities for sine and cosine:\[ \cos(x+y) = \cos x \cos y - \sin x \sin y \]\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]

Step 4: Substitute Angle Identities into Expression

Replace \(\cos(x+y)\) and \(\sin(x+y)\) in the expression for \(\operatorname{cis}(x+y)\):\[ \operatorname{cis}(x+y) = (\cos x \cos y - \sin x \sin y) + i(\sin x \cos y + \cos x \sin y) \]

Step 5: Expand the Right Hand Side

Expand \((\operatorname{cis} \, x)(\operatorname{cis} \, y)\):\[ (\cos x + i\sin x)(\cos y + i\sin y) \]Multiply using the distributive property:\[ = \cos x \cos y + i\cos x \sin y + i\sin x \cos y - i^2\sin x \sin y \]

Step 6: Simplify Using \(i^2 = -1\)

Since \(i^2 = -1\), the expression becomes:\[ \cos x \cos y - \sin x \sin y + i\sin x \cos y + i\cos x \sin y \]

Step 7: Confirm Both Sides are Equal

Now compare both sides:Left-hand side: \(\cos x \cos y - \sin x \sin y + i(\sin x \cos y + \cos x \sin y)\)Right-hand side: \(\cos x \cos y - \sin x \sin y + i\sin x \cos y + i\cos x \sin y\)Both expressions match exactly, confirming the identity is true.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. In this plane, the horizontal axis is the real axis, while the vertical axis is the imaginary axis.
Complex numbers are written as \(a + b \mathbf{i}\), where \(a\) is the real part and \(b\) is the imaginary part. Each complex number can be represented as a point in the plane, with its coordinates \((a, b)\). This makes it simple to visualize addition and multiplication of complex numbers, just like vector operations in two-dimensional geometry.

• Real Axis: Horizontal line for real parts.
• Imaginary Axis: Vertical line for imaginary parts.
• Polar Representation: Uses angle and distance from the origin, represented as \(r(\cos \theta + \mathbf{i} \sin \theta)\).

Unit Circle

The unit circle in the complex plane is a circle with a radius of one, centered at the origin (0,0). It contains all the complex numbers whose absolute value (or modulus) is 1. These numbers are expressed as \( \cos x + \mathbf{i} \sin x \) for some real angle \(x\).

• Equation: \(|z| = 1\), where \(z\) is a complex number.
• Each point on this circle corresponds to an angle \(x\) with the horizontal axis, forming a neat way to connect trigonometric functions and complex numbers.

On the unit circle, the concept of "cis" (short for cosine plus imaginary sine) is used. The notation "cis \(x\)" signifies \( \cos x + \mathbf{i} \sin x \). This compact form is handy when working through complex multiplication and identities.

Angle Sum Identities

Angle sum identities are formulas used to find the cosine and sine of the sum of two angles. This is a key to understanding why \( \operatorname{cis}(x + y) = \operatorname{cis}(x) \cdot \operatorname{cis}(y) \) holds true.

• Sine of Sum: \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
• Cosine of Sum: \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)

These identities allow you to expand the functions \( \cos(x+y) \) and \( \sin(x+y) \) in terms of \( \cos x, \sin x, \cos y, \) and \( \sin y \). Using these, we can show that multiplying two "cis" terms is equivalent to adding their angles. This property is a foundational aspect of how complex numbers and trigonometry overlap.

Trigonometric Functions

Trigonometric functions, like sine and cosine, help describe relationships in triangles and unit circles. In the complex plane, these functions are tied to representing complex numbers in polar form.

• The cosine function corresponds to the adjacent side of the angle in a right triangle over the hypotenuse.
• The sine function corresponds to the opposite side of the angle over the hypotenuse.

In the context of complex numbers, these functions help convert between rectangular and polar forms. Using trigonometric identities, we explore expressions like \( \operatorname{cis} \theta = \cos \theta + \mathbf{i} \sin \theta \). This framework suggests that multiplying complex numbers can be visualized as rotating or scaling these vectors on the complex plane. Thus, these functions not only bridge algebra and geometry but also facilitate understanding the dynamic behaviors of complex number operations.