Physics equations/01-Introduction/A:mathReview

Common misconceptions

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

Percent

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

$\frac{BIG}{SMALL}=1+\underset{percentdifference}{\underbrace{\frac{BIG-SMALL}{SMALL}}}$

Trigonometry

In this right triangle: Template:Nowrap Template:Nowrap Template:Nowrap

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

Logarithms and exponents are inverse functions

$y=b^x⟺x={\log }_{b}y$

The $⟺$ 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:

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

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, $a{x}^2+bx+c=0$, has the solutions:

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

Factoring

If $f(x)\cdot g(x)\cdot h(x)=0$ then $f(x)=0org(x)=0orh(x)=0$

Example:

If $x(x-2)(x-5)=0$ then $x=0orx=2orx=5$