Problem 27 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=9 \operatorname{cis}(\pi) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \(-9\).

Step 1: Understand the cis Notation

The notation \( z = r \operatorname{cis}(\theta) \) is a shorthand for \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument of the complex number. In this case, \( r = 9 \) and \( \theta = \pi \).

Step 2: Apply the Trigonometric Identities

Substitute \( \pi \) for \( \theta \) in the expressions for cosine and sine: \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).

Step 3: Substitute into Rectangular Form

Using the expressions from Step 2, substitute into the rectangular form formula: \[ z = 9(\cos \pi + i \sin \pi) = 9(-1 + i \cdot 0) \] This simplifies to \( z = -9 + 0i \).

Step 4: Express the Result in Standard Rectangular Form

The expression \(-9 + 0i\) can be simplified to simply \(-9\) because the imaginary part is zero.

Key Concepts

cis notation

Cis notation is a convenient way to express a complex number in polar form. When we say a complex number is in the form of \( z = r \operatorname{cis}(\theta) \), it is equivalent to writing \( z = r(\cos \theta + i \sin \theta) \). This shorthand combines the trigonometric functions cosine and sine with the real and imaginary parts of a complex number.
It is especially useful for simplifying multiplication and division of complex numbers since it leverages the properties of exponents.
In our example, where \( r = 9 \) and \( \theta = \pi \), we can see how the cis notation gives us the straightforward replacement for cosine and sine based on known values. Remember: \( \operatorname{cis}(\theta) \) literally stands for \( \cos(\theta) + i \sin(\theta) \).

trigonometric identities

Trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring variables. When dealing with complex numbers in polar form, these identities are a powerful tool.
For instance, in our example, the trigonometric identities we need include \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). By substituting these identities into the expression \( \cos \theta + i \sin \theta \), we simplify solving for the rectangular form.
This simplification is why understanding and using trigonometric identities can greatly aid in efficiently transitioning from cis notation to rectangular form.

complex numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are often written in the form \( a + bi \), where \( a \) is a real number and \( b \) is the imaginary part, with \( i \) being the imaginary unit \( i^2 = -1 \).
Complex numbers are not just for mathematicians; they are used in engineering, physics, and applied mathematics where they facilitate calculations in wave theory, electrical circuits, and more.
Understanding their different forms (rectangular and polar) is crucial because each form presents a different perspective and set of tools for solving mathematical problems. They are key to grasp many advanced mathematical concepts.

modulus and argument

The modulus and argument are fundamental characteristics of complex numbers in polar form. The modulus \( r \) represents the distance from the origin to the complex number in the complex plane, akin to the magnitude of a vector.
The argument \( \theta \) is the direction in which the complex number r points from the positive x-axis, typically measured in radians.
Together, they provide a polar representation of the complex number: \( r(\cos(\theta) + i\sin(\theta)) \).
In the example with \( z = 9 \operatorname{cis}(\pi) \), the modulus is 9 and the argument is \( \pi \). Knowing these allows you to convert easily back and forth between polar and rectangular forms.