Problem 27 - Algebra and Trigonometry

Question

Write each complex number in rectangular form. $$ 4 e^{i \frac{7 \pi}{4}} $$

Step-by-Step Solution

Answer

The rectangular form is 2\( \sqrt{2} \) - 2i\( \sqrt{2} \).

Step 1 - Understand Euler's Formula

Euler's formula states that any complex number in exponential form can be written as \[ z = re^{iθ} = r(\text{cos}(θ) + i \text{sin}(θ)) \] where \( r \) is the magnitude and \( θ \) is the angle.

Step 2 - Identify the Magnitude and Angle

For the given complex number \( 4 e^{i \frac{7 \frac{7 \text{π} }{4}}} \), the magnitude is 4 and the angle is \( \frac{7 \text{π} }{4} \).

Step 3 - Apply Trigonometric Functions

Using the magnitude and angle in Euler's formula, we get: \[ 4 (\text{cos}( \frac{7 \text{π} }{4}) + i \text{sin} ( \frac{7 \text{π} }{4} ))\]

Step 4 - Find the Cosine and Sine Values

Calculate \(\text{cos} ( \frac{7 \text{π} }{4} )= \text{cos} ( - \frac{ \text{π} }{4} ) = \frac { \text{√2} }{2} \) and \( \text{sin} ( \frac{7 \text{π} }{4} ) = \text{sin} (- \frac{ \text{π} }{4} ) = - \frac { \text{√2} }{2} \).

Step 5 - Substitute the Values

Now substitute the cosine and sine values into the equation: \[ 4 \times ( \frac { \text{√2} }{2} +i - \frac { \text{√2} }{2} ) \].

Step 6 - Simplify

Simplify the expression to get the rectangular form: \[ 4 \times \frac { \text{√2} }{2} + 4 \times ( -i \frac { \text{√2} }{2}) = 2 \text{√2} - 2 i \text{√2} \].

Key Concepts

Euler's Formula

Euler's formula is a key concept in understanding complex numbers in exponential form. It states that any complex number can be expressed as \[ z = re^{iθ} = r(\text{cos}(θ) + i \text{sin}(θ)) \]

• \( r \) is the magnitude of the complex number.
• \( θ \) is the angle or argument of the complex number.

This formula connects exponential functions with trigonometric functions, providing a powerful way to manipulate and understand complex numbers. For example, in the given problem, the complex number is written as \( 4 e^{i \frac{7 \text{π}}{4}} \). By using Euler's formula, we can convert it to its rectangular form by calculating the cosine and sine of the angle, and multiplying by the magnitude.

Trigonometric Functions

Trigonometric functions play a crucial role when converting a complex number from its exponential form to its rectangular form. In this context, you'll frequently use the cosine and sine functions from Euler's formula: \( \text{cos}(θ) \) and \( \text{sin}(θ) \) .

• The cosine function gives the real part.
• The sine function gives the imaginary part.

For instance, we have \( \theta = \frac{7 \text{π}}{4} \) in the given exercise. To find the rectangular form, calculate the trigonometric values:
\[ \text{cos}( \frac{7\text{π}}{4} ) = \text{cos}( - \frac{\text{π}}{4} ) = \frac{ \text{√2} }{2} \]
and
\[ \text{sin}( \frac{7\text{π}}{4} ) = \text{sin}( - \frac{\text{π}}{4} ) = - \frac{ \text{√2} }{2} \] . These values are then used to convert the given complex number to its rectangular form.

Magnitude and Angle

Understanding the magnitude and angle is vital when working with complex numbers in exponential form.

• The magnitude \( r \) is the distance from the origin to the point on the complex plane.
• The angle \( θ \) (also known as the argument) indicates the direction of the point relative to the positive real axis.

In the given example, for the complex number \( 4 e^{i \frac{7 \text{π}}{4}} \), we have:

• The magnitude \( r = 4 \).
• The angle \( θ = \frac{7 \text{π}}{4} \).

Using these values, we apply Euler's formula to convert the exponential form to its rectangular form:
\[ 4 (\text{cos}( \frac{7 \text{π}}{4}) + i \text{sin}( \frac{7 \text{π}}{4} )) \] .By substituting the trigonometric values computed earlier, we find:
\[ 4 \bigg( \frac{ \text{√2} }{2} - i \frac{ \text{√2} }{2} \bigg) = 2 \text{√2} - 2i \text{√2} \] .Thus, the rectangular form of the complex number is \( 2 \text{√2} - 2i \text{√2} \).