We can place a point in a plane by polar coordinates Euler's formula states that Employing this formula, we have so we have Cartesian coordinates from the polar coordinates. Therefore, a point in a plane can be represented by polar coordinates which have their equivalent complex numbers
The multiplication of two points and which have their equivalent complex numbers, can be easily calculated. Given two equivalent complex numbers and the multiplication is where the rule of addition of exponents over real numbers is extended to that over complex numbers in the same way. Therefore, the multiplication of two points is where and
What is the multiplication of two points and in polar coordinates?
The first coordinate of the multiplication is the product of the two first coordinates. The second coordinate of the multiplication is the sum of the two second coordinates. Therefore, we have
What is the multiplication of two points in polar coordinates and in Cartesian coordinates? Approximate the angles in radians up to two digits below the decimal point.
Convert in Cartesian coordinates to a representation in polar coordinates:
In polar coordinates, the first coordinate of the multiplication is the product of the two first coordinates, and the second coordinate of the multiplication is the sum of the two second coordinates. Therefore, we have
What is the multiplication of two points and in Cartesian coordinates? Approximate the angles in radians up to two digits below the decimal point.
Convert and in Cartesian coordinates to representations in polar coordinates: and
Therefore, we have
A complex number is represented by the point in the complex plane. From the properties of complex numbers, we can write where . This is shown in the picture below:
Note that the pair can equivalently be described by trigonometric functions of another pair , often denoted . These are referred to as the polar coordinates of the complex number .
is a non-negative number denoting the magnitude of the complex number (the radius of the circle) and is represented on the radial axis that extends outward from the origin at . An expression for can be obtained by dividing by :
and is called the argument of the complex number.
What is the polar form of the complex number
We can draw this point on the complex plane:
We can see from the image that the triangle has two equal sides; from this we know that it must also have two equal angles. Since one of the angles is degrees, we can safely conclude that is equal to degrees or radians. We can then find by computing the hypotenuse of the triangle using the Pythagoras theorem: Thus we can write our complex number as
Write in its polar form.
Applying our formula,
Write in its rectangular form.
We know that and we evaluate Thus our complex number is
in its polar form.
We have Then from our formula, we have Then since we can conclude that
Write in its polar form.
The given expression can be rewritten as Then we have and Thus, the polar form is
If is the polar form of what are and
Observe that can be expressed as where is a positive integer. Then since we can get the value of as follows: Since can be expressed as Thus, since and from trigonometry we can conclude that Therefore, the answer is