(a) The complex number z=-1+i in polar form is z= (b) The...

(a) The complex number $z=-1+i$ in polar form is $z=$__________ (b) The complex number $z=2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$ in rectangular form is $z=$___________ (c) The complex number graphed below can be expressed in rectangular form as___________or in polar form as___________

(a) The complex number $z=-1+i$ in polar form is $z=$__________
(b) The complex number $z=2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$ in rectangular form is $z=$___________
(c) The complex number graphed below can be expressed in rectangular form as___________or in polar form as___________

Key Concepts

Complex Numbers in Rectangular Form

Complex numbers are expressed in rectangular form as a + bi, where 'a' represents the real part and 'b' represents the imaginary part. This format is particularly useful when performing addition and subtraction of complex numbers, as these operations are carried out component-wise.

Complex Numbers in Polar Form

In polar form, a complex number is expressed as r(cos ? + i sin ?), where r is the modulus (or absolute value) and ? is the argument (or angle) of the complex number. This representation is valuable for multiplication, division, and finding powers and roots of complex numbers.

Conversion Between Rectangular and Polar Forms

Conversion between rectangular and polar forms involves computing the modulus and argument from rectangular coordinates, using the formulas r = ?(a² + b²) and ? = arctan(b/a) (with consideration of the correct quadrant), and vice versa by calculating a = r cos ? and b = r sin ?.

Modulus and Argument

The modulus of a complex number is a measure of its distance from the origin in the complex plane, and the argument is the angle it makes with the positive real axis. Together, these components in polar form encapsulate both the magnitude and directional components of the complex number.

Euler's Formula

Euler's formula establishes a deep connection between trigonometric functions and the exponential function, stating that e^(i?) = cos ? + i sin ?. This formula is fundamental in expressing complex numbers in exponential polar form, simplifying many operations such as exponentiation and logarithmic calculations.

Transcript

00:02 So for part a, if we want to convert the complex number into polar form, well, remember, polar form looks like r cosine theta plus isign theta, r times cosine theta plus i sine theta.

00:20 And so we can find the modulus here, r is equal to the square root of 1 plus 1 square.

00:26 So this is r is equal to the square root of 2.

00:29 And so our theta value is theta is equal to tangent inverse of negative 1 over 1 and so this is tangent inverse of negative 1 so we find that theta is equal to 7 pi over 4 7 pi over 4 and so plugging this in gives us the square of 2 times square of 2 times cosine of 7 pi over 4 plus sign of 7 pi over 4 and so so now now or or this this data could also be equal to to three pi over four either either value works and so if we look at part b well we have the the complex value or the complex number two times cosine of pi over six times i sign pi over six and so the pi over six is the squared three over two so we have two times square three over two plus one half times i so this this gives us root 3 plus i and so this is our solution for part b and so now let's look at part c well the graft the graft complex number is as a as a rectangular form of 1 plus i this is the r form so we'll call this rectangular form r form and so if we want to find polar form now what we have to find we have to find this r value so let's call this rec rec form so so so r is equal to the square of two and so now we know that tangent inverse of b over a so this is one is equal to pi over four equal to pi over four and so so if we write this as in polar form this would be square root two times cosine of pi over four times cosine of pi over four plus i sign pi over four plus i sign i sign pi over four.

02:39 And so this is our solution for part b in polar form...

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