User:Cavenba/Mathtastic

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$\pi = 3.141592653589793238462643383279502884197169399375105820974944592307\,\!$

1 Trigonometric functions
- 1.1 Tips
2 Trigonometric equations
- 2.1 Example
3 Trigonometric identities
4 Radians
- 4.1 Important equations
  - 4.1.1 Degrees → Radians
  - 4.1.2 Radians → Degrees

[edit] Trigonometric functions

$\sin \theta = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {o} {h}$

$\cos \theta = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {a} {h}$

$\tan \theta = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {o} {a}$

[edit] Tips

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

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

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

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

[edit] Trigonometric equations

[edit] Example

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

[edit] Trigonometric identities

See also: List of trigonometric identities

The unit circle can be very helpful

[edit] Reciprocal

1. $\csc \theta = \frac {1} {\sin \theta}$

2. $\sec \theta = \frac {1} {\cos \theta}$

3. $\cot \theta = \frac {1} {\tan \theta}$

[edit] Quotient

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

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

[edit] Pythagorean

6. $\sin^2 \theta + \cos^2 \theta = 1\,\!$

7. $\tan^2 \theta + 1 = \sec^2 \theta\,\!$

8. $1 + \cot^2 \theta = \csc^2 \theta\,\!$

[edit] Example

$\mbox{Prove. } \sec x + \frac {1} {\cot x} = \frac {1 + \sin x} {\cos x}$

$\left ( \frac {1} {\cos x} \right ) + \left ( \frac {\sin x} {\cos x} \right ) = \frac {1 + \sin x} {\cos x}$

$\frac {1 + \sin x} {\cos x} = \frac {1 + \sin x} {\cos x}$

[edit] 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.

[edit] Important equations

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