Problem 21 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=6 \operatorname{cis}(0) $$

Step-by-Step Solution

Answer

The rectangular form is \( 6 + 0i \).

Step 1: Understand the 'cis' Function

The notation \( \operatorname{cis}(\theta) \) is shorthand for \( \cos(\theta) + i\sin(\theta) \). Therefore, \( 6 \operatorname{cis}(0) \) becomes \( 6 (\cos(0) + i\sin(0)) \).

Step 2: Evaluate the Trigonometric Functions

Calculate the values of \( \cos(0) \) and \( \sin(0) \). We have \( \cos(0) = 1 \) and \( \sin(0) = 0 \).

Step 3: Substitute and Simplify

Substitute \( \cos(0) = 1 \) and \( \sin(0) = 0 \) into the expression: \( 6(1 + i \cdot 0) \). This simplifies to \( 6 + 0i \).

Step 4: Write in Rectangular Form

The rectangular form of the complex number is expressed with real and imaginary parts, which in this case is \( 6 + 0i \).

Key Concepts

Rectangular Form

Complex numbers can be expressed in different forms, and one of the most common is the rectangular or Cartesian form. This form represents a complex number as a sum of its real and imaginary components, typically written as \( a + bi \). Here, \( a \) is the real part, and \( b \) is the imaginary part.

To convert a complex number from another form into rectangular form, like from polar to rectangular, you simply need to evaluate and express it with its components \( a \) and \( bi \). This form is handy because it makes addition and subtraction straightforward, as you just combine like terms. In our example with the complex number \( 6 \operatorname{cis}(0) \), the rectangular form becomes \( 6 + 0i \).

When a complex number has an imaginary part of zero, such as in \( 6 + 0i \), it is simply a real number, \( 6 \), highlighting the flexibility of the rectangular form.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, play a crucial role in working with complex numbers, especially when converting between forms. These functions help express angles and relationships within the unit circle.

The notation \( \operatorname{cis}(\theta) \) utilizes trigonometric functions to represent cosine and sine together as \( \cos(\theta) + i\sin(\theta) \). This makes it easier to handle the angle \( \theta \) within complex numbers, particularly in polar form.

For example, in the expression \( 6 \operatorname{cis}(0) \):

• The cosine function, \( \cos(0) \), is 1.
• The sine function, \( \sin(0) \), is 0.

Plugging these into the expression highlights the power of trigonometric identities: they enable us to seamlessly convert angles into usable components of complex numbers.

Polar Coordinates

Polar coordinates offer a different perspective to view complex numbers by focusing on their magnitude and angle. This method is especially useful when dealing with multiplication and division of complex numbers, as it leverages the magnitude (or modulus) and the angle (or argument). In polar coordinates, a complex number is represented as \( r \operatorname{cis}(\theta) \), where \( r \) is the magnitude and \( \theta \) is the angle made with the positive x-axis.

Using polar coordinates for complex numbers provides an intuitive understanding—\( r \) tells you how far the number is from the origin, and \( \theta \) tells you the direction. In our case, with \( z = 6 \operatorname{cis}(0) \), \( r \) is 6, and \( \theta \) is 0, indicating a point directly on the positive x-axis at unit distance 6 from the origin.

This visualization helps when plotting complex numbers on the complex plane and simplifies various operations.

• Magnitude \( r = |z| = 6 \)
• Angle \( \theta = 0^\circ \) (pointing right along the x-axis).