Cartesian , Polar and Exponential FormHelp needed thanks .

[mikecrush]

Homework Statement

how can i convert this : - 2 (cos pai / 4 + i sin pai / 4 ) to Cartesian , Polar and Exponential form ?

Homework Equations

z = ( a + i b)

The Attempt at a Solution

r= -2
tan inverse = pai/4 / pai/4
??

Thank you very much for helping me out

 

[HallsofIvy]

It is already almost in "polar form". If you did not see that immediately, you need to review the definitions. The only reason it is not already in polar form is because the "r" in "$r (cos(\theta)+ i sin(\theta))$" cannot be negative. Draw the line with $\theta= \pi/4$ and go backwards: $-2(cos(\theta)+ i sin(\theta))= 2(cos(\theta+ \pi)+ i sin(\theta+ \pi)$

On thing you should know is that the "r" in a polar
To change to "Cartesian form", just evaluate the functions. What is $cos(\pi/4)$? What is $sin(-\pi/4)$? What are $-2 cos(\pi/4)$ and $-2 sin(\pi/4)$?

The "exponential form" of $r(cos(\theta)+ i sin(\theta))$ is $r e^{i\theta}$. Again, r cannnot be negative.

 

[Architeuthis Dux]

with this problem! Converting between different forms of complex numbers, such as Cartesian, Polar, and Exponential forms, is an important skill in the field of mathematics and science. Let's go through each of these forms and see how we can convert the given complex number into each of them.

First, let's start with Cartesian form, which is in the form z = a + ib. To convert the given complex number, -2 (cos π/4 + i sin π/4), into Cartesian form, we need to use the trigonometric identities cos π/4 = √2/2 and sin π/4 = √2/2. This gives us -2 (√2/2 + i√2/2). Simplifying further, we get -√2 - i√2 as the Cartesian form of the given complex number.

Next, let's move on to Polar form, which is in the form z = r(cos θ + i sin θ). Here, r represents the magnitude of the complex number and θ represents the angle in radians. To convert the given complex number into Polar form, we need to use the modulus and argument of the complex number. The modulus, r, is equal to the absolute value of the complex number, which in this case is 2. The argument, θ, can be found using the inverse tangent function, as you have correctly done in your attempt. Therefore, the Polar form of the given complex number is 2(cos π/4 + i sin π/4).

Lastly, let's look at Exponential form, which is in the form z = re^(iθ). To convert the given complex number into Exponential form, we need to use the Euler's formula, e^(iθ) = cos θ + i sin θ. This gives us -2e^(iπ/4). Simplifying further, we get -2(cos π/4 + i sin π/4) as the Exponential form of the given complex number.

I hope this helps you understand how to convert between Cartesian, Polar, and Exponential forms of complex numbers. Keep practicing and you will become more comfortable with these conversions. Good luck with your homework!