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

${\mbox{Remember: SOHCAHTOA}}\,\!$

$\sin \theta =o/h{\mbox{ or SOH}}\,\!$

$\cos \theta =a/h{\mbox{ or CAH}}\,\!$

$\tan \theta =o/a{\mbox{ or TOA}}\,\!$

Trigonometric equations

Example

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

Trigonometric identities

See also: List of trigonometric identities

The unit circle can be very helpful

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

An angle of 1 radian subtends an arc equal in length to the radius of the circle.

See also: Radian

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

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