Question

Write each complex number in rectangular form. $$ 3 e^{i \frac{\pi}{2}} $$

Short answer

The rectangular form is \( 3i \).

Step-by-step solution

- Understand Polar and Rectangular Forms

Complex numbers can be expressed in polar form as \( r e^{i \theta} \) and in rectangular form as \( a + bi \). In this problem, the given complex number is in polar form \( 3 e^{i \frac{\pi}{2}} \).

- Identify the Magnitude and Angle

In the given polar form, \( r = 3 \) (the magnitude) and \( \theta = \frac{\pi}{2} \) (the angle).

- Use Euler's Formula

Euler's formula states that \( e^{i \theta} = \cos(\theta) + i \sin(\theta) \). Substitute \( \theta = \frac{\pi}{2} \) into this formula to get \( e^{i \frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \).

- Simplify Trigonometric Functions

Evaluate the cosine and sine functions for \( \theta = \frac{\pi}{2} \): \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \). Therefore, \( e^{i \frac{\pi}{2}} = 0 + i(1) = i \).

- Multiply by the Magnitude

Now, multiply the result by the magnitude \( r = 3 \): \( 3 \cdot i = 3i \).

Final Answer

Putting it all together, the complex number \( 3 e^{i \frac{\pi}{2}} \) in rectangular form is \( 0 + 3i \).

Key concepts

Polar Form

In mathematics, complex numbers can be represented in different forms. The polar form is useful when working with trigonometric and exponential functions. This form expresses a complex number as \( r e^{i \theta} \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument). Polar form is particularly handy for multiplication, division, and finding powers of complex numbers.

The magnitude \( r \) is the distance from the origin to the point in the complex plane, computed as \( r = \sqrt{a^2 + b^2} \). The angle \( \theta \), also known as the argument, represents the direction of the line connecting the origin to the point. It can be calculated using trigonometric relationships.

For example, in the exercise given, \( 3 e^{i \frac{\pi}{2}} \), the magnitude \( r \) is 3, and the angle \( \theta \) is \frac{\pi}{2}\.

Euler's Formula

Euler's formula is a fundamental bridge between the exponential function and trigonometric functions. It states that \( e^{i \theta} = \cos(\theta) + i \sin(\theta) \). This formula is a key concept in complex analysis and signal processing.

In the context of the exercise, we use Euler's formula to convert polar form to rectangular form. Given \( \theta = \frac{\pi}{2} \), substitute it into Euler's formula:

\[ e^{i \frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \]

By simplifying the trigonometric functions, we find that \cos(\frac{\pi}{2}) = 0 \ and \sin(\frac{\pi}{2}) = 1 \, so:
\[ e^{i \frac{\pi}{2}} = 0 + i(1) = i \]

This shows how Euler's formula helps convert the exponential part of the polar form into its corresponding sine and cosine components in rectangular form.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential in various branches of mathematics. In the context of complex numbers, they play a crucial role in converting between different forms.

In the provided example, to convert from polar to rectangular form, we need to evaluate trigonometric functions at specific angles. For the angle \ \frac{\pi}{2} \, we found:

• \ \cos(\frac{\pi}{2}) = 0 \

• \ \sin(\frac{\pi}{2}) = 1 \


This provides us the necessary components to rewrite the complex number. Once we have \( e^{i \frac{\pi}{2}} \) in its rectangular equivalent \( 0 + i \)
,
we then multiply by the magnitude \( r = 3 \):

\3 \cdot i = 3i
This final step gives us the rectangular form of the complex number, showing how trigonometric functions simplify this transformation process.