User:Cavenba/Mathtastic

$$\pi = 3.141592653589793238462643383279502884197169399375105820974944592307\,\!$$

Trigonometric functions
$$\sin \theta = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {o} {h}$$ $$\cos \theta = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {a} {h}$$ $$\tan \theta = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {o} {a}$$

Tips
$$ \mbox{Remember: SOHCAHTOA}\,\!$$ $$ \sin \theta = o/h \mbox{ or SOH}\,\!$$ $$ \cos \theta = a/h \mbox{ or CAH}\,\!$$ $$ \tan \theta = o/a \mbox{ or TOA}\,\!$$

Example
$$\mbox{If } \sin x = \frac {\sqrt {3}} {2}$$ $$\mbox{then } x = \begin{cases} 60^\circ + 360^\circ k, k \in \mathbb{I} \\ 120^\circ + 360^\circ k, k \in \mathbb{I} \end{cases} $$

Reciprocal
1. $$\csc \theta = \frac {1} {\sin \theta}$$ 2. $$\sec \theta = \frac {1} {\cos \theta}$$ 3. $$\cot \theta = \frac {1} {\tan \theta}$$

Quotient
4. $$\tan \theta = \frac {\sin \theta} {\cos \theta}$$ 5. $$\cot \theta = \frac {\cos \theta} {\sin \theta}$$

Pythagorean
6. $$\sin^2 \theta + \cos^2 \theta = 1\,\!$$ 7. $$\tan^2 \theta + 1 = \sec^2 \theta\,\!$$ 8. $$1 + \cot^2 \theta = \csc^2 \theta\,\!$$

Example
$$\mbox{Prove. } \sec x + \frac {1} {\cot x} = \frac {1 + \sin x} {\cos x}$$ $$\left ( \frac {1} {\cos x} \right ) + \left ( \frac {\sin x} {\cos x} \right ) = \frac {1 + \sin x} {\cos x}$$ $$ \frac {1 + \sin x} {\cos x} = \frac {1 + \sin x} {\cos x}$$

Radians


The radian is a unit of plane angle, equal to 180/π degrees, or about 57.296 degrees. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level.

Important equations
$$\mbox{Arc Length} = 2 \pi r \left (\frac {\theta} {360^\circ} \right )$$ $$1 \mbox{ complete circle} = 360^\circ \left (\frac {1 \mbox{ rad}} {57.296^\circ} \right ) = 2\pi$$

Degrees → Radians
$$30^\circ \left (\frac {\pi} {180^\circ} \right) = \frac {30 \pi} {180} = \frac {\pi} {6} \mbox{ rad}$$

Radians → Degrees
$$\frac {3 \pi} {4} \left (\frac {180^\circ} {\pi} \right ) = 135^\circ$$