What is the Complex Angle for 2√(3)-2i?

[guyttt]

Homework Statement

I need to find the complex angle θ for: 2√(3)-2i in polar form.

Homework Equations

The Attempt at a Solution

If I draw a complex plane, I can see that 2√3 on the real axis gives 0°, and -2i gives 3π/2 (270°), but it's incorrect. How can I find the complex angle of 2√(3)-2i?

Thanks in advance.

 

[HallsofIvy]

The complex number 2\sqrt{3}- 2i corresponds to the point (2\sqrt{3}, -2) in the complex plane. Its "argument" is the angle the line from the origin, (0, 0), makes with the positive real axis. In general the "angle", or "argument", of the complex number a+ bi, is given by tan^{-1}\left(\frac{b}{a}\right).

What is tan^{-1}\left(\frac{1}{\sqrt{3}}\right)?

(That should not be called a "complex angle". A "complex angle" would be a complex number, \theta, such that, say, sin(\theta)= 2\sqrt{3}- 2i.)

 

[guyttt]

The complex number 2\sqrt{3}- 2i corresponds to the point (2\sqrt{3}, -2) in the complex plane. Its "argument" is the angle the line from the origin, (0, 0), makes with the positive real axis. In general the "angle", or "argument", of the complex number a+ bi, is given by tan^{-1}\left(\frac{b}{a}\right).

What is tan^{-1}\left(\frac{1}{\sqrt{3}}\right)?

(That should not be called a "complex angle". A "complex angle" would be a complex number, \theta, such that, say, sin(\theta)= 2\sqrt{3}- 2i.)

Hey, thank you for replying.

I understand that you can find the angle using tan−1, but I have to find the argument without using a calculator. Maybe it is obvious what angle tan−1(1/√3) gives, but I haven't worked with complex planes in a long time so it's not really that clear for me. Is there any way I can find the angle without using tan−1?

Thanks again!

 

[HallsofIvy]

There are some angles for which it is easy to find the trig function values. For example, the angle 45 degrees (\pi/4 radians). If one angle of a right triangle is 45 degrees then the other must be also so this is an isosceles triangle. Taking the legs to have length 1, by the Pythagorean theorem, the hypotenuse has length \sqrt{2} and all of the trig values can be written down.

Consider an equilateral triangle. It has all three sides of the same length (say, 1) and all three angles 60 degrees (\pi/3 radians). Drawing a line from one vertex perpendicular to the opposite side divides it into two right triangles having angles 60 and 30 degrees (\pi/6 radians) with hypotenuse of length 1 and one leg of length 1/2. By the Pythagorean theorem, the other leg has length \sqrt{1- 1/4}= \sqrt{3/4}= \sqrt{3}/2. All trig functions, of both 30 degrees and 60 degrees can be calculated from that.

 

[SteamKing]

Hey, thank you for replying.

I understand that you can find the angle using tan−1, but I have to find the argument without using a calculator. Maybe it is obvious what angle tan−1(1/√3) gives, but I haven't worked with complex planes in a long time so it's not really that clear for me. Is there any way I can find the angle without using tan−1?

Thanks again!

This graphic gives you the information about complex numbers you need:

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j is used by electrical engineers in place of i. i ² = j ² = -1