Write the complex number in polar form. Express the argument in radians. -5+5squareroot3 i

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Answer 1

In order to convert to polar form, remember that

 

x + yi = r[cos(a) + isin(a)],

 

in which r, your magnitude, is sqrt[x^2 + y^2]

and the reference angle of your argument, a, is arctan(|y/x|) (i.e., tan inverse of the absolute value of y/x... using absolute value gives us a positive acute angle, which is our reference angle).

 

Then, the signs of x & y in the original determines the quadrant (and, thereby, the actual value) of your argument

 

r = sqrt[(-5)^2 + (5sqrt3)^2] = sqrt[25 + 75] = sqrt[100] = 10

 

reference angle of a = arctan(|5sqrt3/-5|) = arctan(sqrt3)... Remember, arctan(sqrt3) is the angle whose tangent is sqrt3. Tan of 60degrees (or pi/3 radians) is sqrt3. So, reference angle is 60degrees or pi/3 radians.

 

Since, x is negative and y is positive, our quadrant is the second; our actual argument becomes 180 - 60 degrees (or pi - pi/3 radians)... This simplifies to 120 degrees or 2pi/3 radians.

 

So, polar form is 10(cos(120degrees) + isin(120degrees)) or  

10(cos(2pi/3radians) + isin(2pi/3radians)).