Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. \(25 e^{2 i} \quad\) Carefull The angle is 2 radians.

Short answer

\( -10.4025 + 22.7325i \)

Step-by-step solution

- Identify the components

The given expression is in the polar form: \[ 25 e^{2i} \]Here, the magnitude \( r \) is 25 and the angle \( \theta \) is 2 radians.

- Convert from polar to rectangular form

To convert from polar to rectangular form, use the equations:\[ x = r \, \text{cos}( \theta ) \] and \[ y = r \, \text{sin}( \theta ) \] Substitute \( r = 25 \) and \( \theta = 2 \):\[ x = 25 \, \text{cos}(2) \] \[ y = 25 \, \text{sin}(2) \] Using a calculator, find the values of cos(2) and sin(2):\[ x = 25 \, (-0.4161) = -10.4025 \] \[ y = 25 \, (0.9093) = 22.7325 \]

- Write the number in rectangular form

Combine \( x \) and \( y \) to write the number in the form \( x + iy \):\[ -10.4025 + 22.7325i \]

- Plot the number in the complex plane

On the complex plane, plot the point \( -10.4025 + 22.7325i \). The x-coordinate is -10.4025 and the y-coordinate is 22.7325.

Key concepts

polar form

Complex numbers can be represented using different forms, and one of these is the polar form. This form expresses a complex number in terms of its magnitude and angle, making it quite useful for multiplication and division. Polar form is written as: \[ r e^{i \theta} \]Here, \( r \) is the magnitude (or distance from the origin), and \( \theta \) is the angle (measured in radians) from the positive real axis. To convert a number to polar form, follow these steps:

• Determine the distance from the origin, also known as the modulus, \( r = |z| \).
• Find the angle \( \theta \), also known as the argument, using the inverse tangent function: \( \theta = \text{arg}(z) = \text{tan}^{-1}(y/x) \).

The polar form makes it easier to handle multiplication and division, as you can simply add or subtract the angles and multiply or divide the magnitudes. In the exercise, we started with the polar form: \( 25 e^{2i} \). The magnitude is 25, and the angle is 2 radians.

rectangular form

The rectangular (or Cartesian) form represents a complex number using its real and imaginary parts. It is written as:\[ x + iy \]Here, \( x \) is the real part and \( y \) is the imaginary part. To convert between polar and rectangular forms, we use trigonometric functions. For our exercise, we converted polar form \( 25 e^{2i} \) into rectangular form by evaluating:\[ x = r \text{ cos}(\theta) \]\[ y = r \text{ sin}(\theta) \]Substituting \( r = 25 \) and \( \theta = 2 \), we get:\[ x = 25 \text{ cos}(2) \approx 25 \times (-0.4161) = -10.4025 \]\[ y = 25 \text{ sin}(2) \approx 25 \times 0.9093 = 22.7325 \]Hence, the rectangular form simplifies to \( -10.4025 + 22.7325i \). This form is useful when adding or subtracting complex numbers, as you can directly combine the real and imaginary components.

complex plane

The complex plane is a two-dimensional plane where complex numbers are visually represented. The horizontal axis corresponds to the real part, and the vertical axis corresponds to the imaginary part. Here’s how to plot a complex number on the complex plane:

• The real part \( x \) determines the horizontal position.
• The imaginary part \( y \) determines the vertical position.

In our exercise, we plotted the number \( -10.4025 + 22.7325i \). To do this:

• Move \( -10.4025 \) units to the left along the real axis.
• Then move \( 22.7325 \) units up along the imaginary axis.

The resulting point is the graphical representation of the complex number on the plane. This visual method helps us understand the position and magnitude of complex numbers better.