66.7: Mathematical Subtleties

• When taking the square root of both sides of an equation, a $\pm$ sign must always be introduced. For example:

$$x^{2}=a \quad \Rightarrow \quad x= \pm \sqrt{a}$$

Both roots may be valid, or, depending on the problem, it may be that one root or the other may be rejected on mathematical or physical grounds.

• Dividing an equation through by a variable may result in losing roots. For example, suppose we have

$$x^{2}-a x=0$$

Dividing through by the variable $x$ will result in one solution, $x=a$; the solution $x=0$ has been lost. Instead of dividing through by the variable $x$, the proper procedure is to factor out an $x$ :

$$x(x-a)=0$$

Since the product on the left-hand side is zero, it follows that either $x=0$ or $x-a=0$, and we retain both roots.

• The relation

$$\sqrt{x} \sqrt{y}=\sqrt{x y}$$

is valid only for $x, y \geq 0$.

• Some mathematical conventions:
  • $1$ is not considered a prime number.
  • $0 !=1$
  • $0^{0}=1$

• When taking an inverse trigonometric function, there will in general be two correct values; your calculator will give only one value, the principal value (P.V.). The other value is found using the table below.

Function	P.V.	Other value
$\arcsin$	$\theta$	$\pi-\theta$
$\arccos$	$\theta$	$-\theta$
$\arctan$	$\theta$	$\pi+\theta$
$\operatorname{arcsec}$	$\theta$	$-\theta$
$\operatorname{arccsc}$	$\theta$	$\pi-\theta$
$\operatorname{arccot}$	$\theta$	$\pi+\theta$

• For arctan $(y / x)$, add $\pi$ to the calculator's principal value answer if $x<0$.