Squares Erase the Signs

A point on the circle of radius 1 is $\frac{3}{5}$ high. How far across is it? Most people say $\frac{4}{5}$. That is only half the answer. A second point at the same height is on the other side.

Subtract ${\sin\left(x\right)}^{2}$ from both sides: ${\cos\left(x\right)}^{2} = 1-{\sin\left(x\right)}^{2} = 1-\frac{9}{25}$, which is $\frac{16}{25}$. Two numbers have the square 16/25: $\frac{4}{5}$ and $\frac{-4}{5}$. The identity gives the size. The quadrant gives the sign. In quadrant II, left of the y-axis, $\cos\left(x\right) = \frac{-4}{5}$.