Question

Express the given complex number in the form \(R(\cos \theta+\) \(i \sin \theta)=R e^{i \theta}\) $$ -1-i $$

Short answer

Question: Express the complex number \(-1-i\) in the form \(R(\cos \theta + i \sin \theta)=R e^{i \theta}\). Answer: \(\sqrt{2} e^{i\frac{5\pi}{4}}\)

Step-by-step solution

Find the magnitude (R)

To find the magnitude of \(-1-i\), calculate the square root of the sum of the squares of the real and imaginary parts. $$ R = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} $$

Find the argument θ

To find the angle (argument) θ, use the atan2 function, which gives values in the correct quadrant. The atan2 function takes the ratio of the imaginary part to the real part as input. $$ \theta = \operatorname{atan2}(-1,-1) = \frac{5\pi}{4} $$

Express the complex number in polar form

Now that we have the magnitude (R) and the argument (angle θ), we can express the complex number in polar form as: $$ R(\cos \theta + i \sin \theta)=\sqrt{2}(\cos\frac{5 \pi}{4} + i\sin\frac{5 \pi}{4}) $$

Express the complex number in exponential form

Finally, we can express the complex number in exponential form using Euler's formula, \(Re^{i\theta}\) : $$ \sqrt{2} e^{i\frac{5\pi}{4}} $$ Thus, the given complex number \(-1-i\) can be expressed as \(\sqrt{2} e^{i\frac{5\pi}{4}}\).

Key concepts

Polar Form

Complex numbers can be represented in multiple ways, one of which is the polar form. The polar form expresses a complex number using its magnitude and direction, much like how you might describe a vector.
For a complex number in the form of polar coordinates, you will see it written as \( R(\cos \theta + i\sin \theta) \), where:

• \( R \): The magnitude of the complex number.
• \( \theta \): The angle (in radians) that the line makes with the positive real axis, also known as the argument.

In the given exercise, the complex number \(-1-i\) is expressed in polar form as \( \sqrt{2}(\cos \frac{5\pi}{4} + i\sin\frac{5\pi}{4}) \). This indicates a magnitude of \( \sqrt{2} \) and an angle of \( \frac{5\pi}{4} \). Recognizing this form can be helpful especially in multiplying complex numbers, as it's often easier to handle angles than handling separate real and imaginary components.

Magnitude and Argument

The magnitude and argument of a complex number provide both a sense of scale and direction on the complex plane.
The magnitude, \( R \), gives you the distance from the origin to the point in the complex plane, calculated using the Pythagorean theorem. For \(-1-i\), it was computed as \( \sqrt{2} \). This involves adding the squares of the real part and the imaginary part and then taking the square root of the result.
The argument, \( \theta \), represents the angle made with the positive x-axis, which shows the direction of the complex number. One effective method for finding the argument is by utilizing the \( \operatorname{atan2} \) function, which correctly considers the sign of both parts to place the angle in the correct quadrant. For \(-1-i\), it gives an angle of \( \frac{5\pi}{4} \).

• **Magnitude (\( R \)):** The "length" of the vector representing the complex number.
• **Argument (\( \theta \)):** Also known as the angle or phase, guides the "direction" of the vector.

Understanding both parameters is key to converting and working with complex numbers in different forms.

Exponential Form

The exponential form of complex numbers is a powerful way to represent them, leveraging Euler's formula, which links exponentials to trigonometric functions. The exponential form of a complex number \( z = R e^{i\theta} \) makes it straightforward to perform many mathematical operations, like multiplication or finding powers and roots of complex numbers.
Using Euler's formula \( e^{i\theta} = \cos \theta + i\sin \theta \), you can realize that this form efficiently marries the magnitude (\( R \)) of the complex number with its angle (\( \theta \)). For the complex number \(-1-i\), the exponential form is \( \sqrt{2}e^{i\frac{5\pi}{4}} \).

• Exponential form encapsulates both components (magnitude and argument) in a compact expression.
• It's highly useful in advanced operations and is often preferred in fields such as electrical engineering and physics.

Choosing exponential form can simplify calculations, and it offers a neat symmetry that blends algebra and geometry seamlessly.