Converting a number to rectangular form

In summary, the rectangular form of the complex number z=a+bi with modulus 2 and principal argument theta = -pi/6 is 2cos(-pi/6) + i2sin(-pi/6). This is obtained by using Euler's formula, which states that the exponential form of a complex number with modulus r and argument theta is r*exp(i*theta)=r*(cos(theta)+i*sin(theta)). To get the length of the real part of z, we can use the equation x=r*cos(theta), which in this case gives us x=2*cos(-pi/6). To get the value of x in decimal form, we can use the method of drawing an equilateral triangle with side 1,
  • #1
Ry122
565
2
Find the rectangular form of the complex number z=a+bi which has modulus 2 and principal argument theta = -pie/6
2cos(-pie/6) to get length of x
=sqrt3 What is the method used to get an answer in this form instead of decimal form (1.732) ?
 
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  • #2
Use Euler's formula
 
  • #3
Could you please show me the steps?
 
  • #4
The complex number with modulus r and argument theta is r*exp(i*theta)=r*(cos(theta)+i*sin(theta)). exp(i*theta)=cos(theta)+i*sin(theta) is Euler's formula.
 
  • #5
Isn't there a quicker way to do it?
 
  • #6
?? If you followed that then the real part of z is x=r*cos(theta)=2*cos(-pi/6). What do you mean 'quicker'. I just wrote it down. I didn't compute anything. How could anything else be quicker?
 
  • #7
I have that in my initial post. I was just wanting to know how u get sqrt3 instead of 1.732.
 
  • #8
Oh. Sorry. Draw an equilateral triangle with side 1. Then pi/6 is half of one of the angles (since they are all pi/3). So split the equilateral triangle into two right triangles with base 1/2 and hypotenuse 1. The remaining side is sqrt(3)/2. Use pythagoras. cos(pi/6) is that remaining side over the hypotenuse. So cos(pi/6)=sqrt(3)/2. So 2*cos(pi/6)=sqrt(3). Misunderstood your question.
 

1. What is rectangular form?

Rectangular form is a way of representing complex numbers using two real numbers, the real part and the imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part multiplied by the imaginary unit i.

2. How do I convert a number to rectangular form?

To convert a number to rectangular form, you first need to identify the real and imaginary parts of the number. The real part is the number without the i and the imaginary part is the number multiplied by i. Then, you can combine the real and imaginary parts with a plus sign in between to get the rectangular form of the number.

3. What is the difference between rectangular form and polar form?

The main difference between rectangular form and polar form is the way they represent complex numbers. Rectangular form uses the real and imaginary parts, while polar form uses the modulus and argument of the complex number. The modulus is the distance from the origin to the complex number, and the argument is the angle between the positive real axis and the vector connecting the origin and the complex number.

4. Can all complex numbers be represented in rectangular form?

Yes, all complex numbers can be represented in rectangular form. This is because rectangular form is a standard and universal way of representing complex numbers, and it can accurately represent both real and imaginary numbers.

5. How is rectangular form used in mathematics and science?

Rectangular form is used in mathematics and science to perform calculations with complex numbers. It is also used to graph complex numbers on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. This allows for a visual representation of complex numbers and their operations, making it a useful tool in various fields of study.

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