Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 5(\cos \pi+i \sin \pi) $$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(-5\).

Step 1: Identify the expression

The expression given is \(5(\cos \pi + i \sin \pi)\). This is a complex number in polar form.

Step 2: Evaluate trigonometric functions

Calculate \(\cos \pi\) and \(\sin \pi\). We know that \(\cos \pi = -1\) and \(\sin \pi = 0\).

Step 3: Substitute values into the expression

Replace \(\cos \pi\) with \(-1\) and \(\sin \pi\) with \(0\) in the expression: \[5(-1 + i \, 0)\].

Step 4: Simplify the expression

Distribute the 5 in the expression: \[5(-1) + 5(0)i = -5 + 0i\].

Step 5: Write the expression in the form a + bi

The expression \(-5 + 0i\) can be simplified to \(-5\), where \(a = -5\) and \(b = 0\).

Key Concepts

Polar Form

Complex numbers can be expressed in two main forms: Cartesian and polar. Polar form is a way to represent complex numbers using their magnitude and angle. The general structure of a complex number in polar form is given by:

• A modulus (or magnitude) usually denoted as \( r \).
• An argument (or angle) usually denoted as \( \theta \).

This is written in the form \( r (\cos \theta + i \sin \theta) \). The modulus \( r \) measures the distance of the point from the origin in the complex plane, while the argument \( \theta \) measures the direction or angle the point makes with the positive real axis.

In the exercise you're provided, the expression \(5(\cos \pi + i \sin \pi)\) represents a complex number where \( r = 5 \) and \( \theta = \pi \). The use of trigonometric terms \( \cos \) and \( i \sin \) signifies its polar nature.

Trigonometric Functions

Trigonometric functions such as \( \cos \) (cosine) and \( \sin \) (sine) are vital when dealing with complex numbers in polar form. These functions help define the position of a point on the unit circle based on a given angle.

- \( \cos \theta \) corresponds to the adjacent side of a right triangle formed by the angle \( \theta \) on the unit circle.- \( \sin \theta \) corresponds to the opposite side.

For the angle \( \pi \), we find \( \cos \pi = -1 \) and \( \sin \pi = 0 \). These values are crucial in transforming a number in polar form into its equivalent in the standard form \(a + bi\).

In the given problem, by identifying these trigonometric values, we substitute \( \cos \pi \) and \( \sin \pi \) into the expression to convert it into a more familiar format of a complex number.

Real Numbers

Real numbers are a key component in representing complex numbers in Cartesian form. A complex number \( a + bi \) is composed of:

• The real part, represented by \( a \), which lies along the x-axis in the complex plane.
• The imaginary part, represented by \( b \), multiplied with \( i \), which lies along the y-axis.

While complex numbers are often used in equations, operations, and transformations, the real numbers specifically are values that can be positive, negative, or zero without an imaginary component.

In the provided exercise, the final result of the operation \(-5 + 0i\) can be written simply as \(-5\), where \(-5\) is an entirely real number. Thus, understanding that \(-5\) aligns with the concept of real numbers is crucial to fully grasping the conversion from polar to standard form.