Physics equations/01-Introduction/A:mathReview

Common misconceptions

$\left({\frac {1}{x}}+{\frac {1}{y}}\right)^{-1}\neq x+y$   and   ${\sqrt {a^{2}+b^{2}}}\neq a+b$.

Percent

The $X\%$ symbol means $X/100$. A quick and dirty way to find the percent difference is to divide the big number by the small:

${\frac {BIG}{SMALL}}=1+\underbrace {\frac {BIG-SMALL}{SMALL}} _{percent\;difference}$

Trigonometry

In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.

$\sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{\,c\,}}\,.$   $\cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{\,c\,}}\,.$   $\tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{\,b\,}}={\frac {\sin A}{\cos A}}\,.$

Logarithms and exponents are inverse functions

$y=b^{x}\iff x=\log _{b}y$

The $\iff$ implies that the statements are equivalent.

The three most common bases are $b=2,e,10$.

The natural log is defined as $\ln y\equiv \log _{e}y$.

If $f=f(x)$ and $g=g(y)$ are inverse functions, then:

$g(f(x))=x$ and $f(g(y))=y$, and we write:

$f=g^{-1}$ and $g=f^{-1}$.

Warning: Do not be confused about this notations. The inverses are NOT multiplicative inverses:

$f^{-1}\neq {\frac {1}{f}}$.

Complexities occur when the inverse is not a true function, or equivalently, when the inverse is multi-valued:

$\tan ^{-1}(\tan \theta )=\theta \;or\;\theta +\pi$

Here the problem arises because,

$\tan(\theta )=\tan(\theta +\pi )$,

so that knowing the tangent of angle does not precisely tell you what the angle was.

$\tan ^{-1}$ is called the 'arctangent', or the 'inverse tangent'. $\sin ^{-1}$ is called 'arcsine', or the 'inverse sine' and so forth.

Quadratic equation

This quadratic equation, $ax^{2}+bx+c=0$, has the solutions:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$

Factoring

If $f(x)\cdot g(x)\cdot h(x)=0$ then $f(x)=0\;or\;g(x)=0\;or\;h(x)=0$

Example:

If $x(x-2)(x-5)=0$ then $x=0\;or\;x=2\;or\;x=5$