How did theta become equal to 3pi/4 here?

Question

How did theta become equal to 3π/4 in this particular example?

Find a set of polar coordinates (r,θ) of the cartesian point (-4,4) such that -2π ≤ θ ≤ 2π and

a. r > 0 and θ > 0 b. r > 0 and θ < 0 c. r < 0 and θ > 0
d. r < 0 and θ < 0

Solution:

x² + y² = r²

r = ±√x² + y² = ±√32 = ±4√2

tanθ = y/x

tanθ = -4/4 = -1 => θ = 3π/4, since (-4,4) ε QII

Answer 1

We want a number (angle) in the interval $[0,2\pi)$ whose cosine is $\frac{-4}{4\sqrt{2}}$ and whose sine is $\frac{4}{4\sqrt{2}}$. There is only one. Cosine negative and sine positive puts us in the second quadrant.

Note that it is not enough to look at the tangent function.