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 \(\sqrt{2} e^{i\frac{5\pi}{4}}\).