Physics equations/01-Introduction/A:mathReview

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

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

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:


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

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, $$ax^2 + bx + c = 0$$, 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 $$x=0\;or\;x=2\;or\;x=5$$