Stephanie M. answered • 05/18/15

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4 - 3i means you've gone 4 right and 3 down from the origin. That makes a right triangle with two legs of lengths 3 and 4 and a hypotenuse of length of r (the length of the vector from the origin to (4, -3)). Solve for r using the Pythagorean Theorem:

c

^{2}= a^{2}+ b^{2}r

^{2}= 3^{2}+ 4^{2}r

^{2}= 9 + 16r

^{2}= 25r = 5

Now, let's find the value of θ, the angle described by that vector. Since θ is in Quadrant IV, tangent will be negative.

tan(θ) = opposite/adjacent = -3/4

θ = tan

^{-1}(3/4) = -36.9°That's not between 0 and 360, so add 360 to find an equivalent angle:

θ = -36.9° + 360° = 323.1°

One way to write this in Polar Form is 5 ∠323.1°.