# Question #07ebb

Aug 13, 2016

Invalid as a general formula..

#### Explanation:

The principal value from the calculator for $\arctan \left(\frac{y}{x}\right) \in Q 1$

when $\left(x , y\right) \in Q 1 \mathmr{and} Q 3$.

Likewise,

the principal value from the calculator for

$a r c \tan \left(\frac{y}{x}\right) \in Q 2$, when $\left(x , y\right) \in Q 2 \mathmr{and} Q 4$

Test with $\left(x , y\right) = \left(\pm 1 , \pm 1\right)$

Any formula equivalent to the couple

$' \cos \theta = \frac{x}{r} \mathmr{and} \sin \theta = \frac{y}{r} '$, for fixing $\theta$, is valid.

As sine and cosine are periodic with period $2 \pi$, $\theta$ identified

in $\left[0 , 2 \pi\right]$ would fix the quadrant correctly.
.