Find in the form, ##x+iy## in the given complex number problem

In summary, to find the value of ##z##, we can expand the expression and match the real and imaginary parts. It is not necessary to convert from radians to degrees and it is helpful to know basic identities and relationships, such as the angle sum and difference formulae.
  • #1
chwala
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Homework Statement
Find, in the form ##x+iy##, the complex numbers given in the polar coordinate form by;

##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]##
Relevant Equations
complex numbers
This is the question as it appears on the pdf. copy;

1682349943342.png


##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]##

My approach;

##\dfrac{3π}{4}=135^0##

##\tan 135^0=-\tan 45^0=\dfrac{-\sqrt{2}}{\sqrt{2}}##

therefore,

##z=-\sqrt{2}+\sqrt{2}i##

There may be a better approach.
 
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  • #2
It should not be necessary to convert from radians to degrees. One should know [tex]
\begin{split}
\cos 0 &= \sin \frac \pi 2 = 1 \\
\cos \frac \pi 6 &= \sin \frac \pi 3 = \frac{\sqrt{3}}2 \\
\cos \frac \pi 4 &= \sin \frac \pi 4 = \frac 1{\sqrt{2}} \\
\cos \frac \pi 3 &= \sin \frac \pi 6 = \frac 12 \\
\cos \frac \pi 2 &= \sin 0 = 0 \end{split}[/tex] These, together with basic identities such as the angle sum and difference formulae, suffice to answer these questions.
 
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  • #3
chwala said:
##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]##
Expanding gives ##z=2\cos (\frac{3π}{4}) + 2 i \sin (\frac{3π}{4})##.

So it’s simply a case of matching the real and imaginary parts:
##x = 2\cos ( \frac{3π}{4})##
##y = 2 \sin ( \frac{3π}{4})##

No need to use ‘##\tan##’. And as already noted by @pasmith, it’s worth getting used to working in radians.

A couple of useful relationships are
##cos(\frac π2 + θ) = - sinθ## and
##sin(\frac π2 + θ) = cos(θ)##.
For example, using the first relationship tells you ##\cos (\frac{3π}{4}) = -sin ( \frac π4)##.
 
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1. What is the purpose of finding ##x+iy## in a complex number problem?

The purpose of finding ##x+iy## in a complex number problem is to represent the complex number in its standard form, also known as the rectangular form. This form makes it easier to perform mathematical operations and to graph the complex number on a Cartesian plane.

2. How do I find ##x+iy## in a complex number problem?

To find ##x+iy## in a complex number problem, you need to identify the real part (x) and the imaginary part (iy) of the complex number. The real part is the coefficient of the term without the imaginary unit (i), and the imaginary part is the coefficient of the term with the imaginary unit. Then, you can simply write the complex number in the form ##x+iy##.

3. Can I write a complex number in the form ##x+iy## if it already has that form?

Yes, you can write a complex number in the form ##x+iy## even if it already has that form. This will not change the value of the complex number, but it can make it easier to work with and understand.

4. Is there a difference between the form ##x+iy## and the polar form of a complex number?

Yes, there is a difference between the form ##x+iy## and the polar form of a complex number. The polar form uses the modulus (distance from the origin) and the argument (angle from the positive real axis) to represent a complex number, while the form ##x+iy## uses the real and imaginary parts to represent a complex number.

5. How can I use the form ##x+iy## to graph a complex number?

To graph a complex number using the form ##x+iy##, you can plot the real part (x) on the horizontal axis and the imaginary part (y) on the vertical axis. This will give you the coordinates of the complex number on a Cartesian plane. You can also use this form to perform translations, rotations, and reflections on the graph of the complex number.

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