Trigonometry

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Circle

Trigonometry unit circle.png
\( (\color{blue}{\cos} \: \theta ,\color{red}{\sin}\: \theta) \)

Radians

Trigonometry radian definition circle.png
Degrees Radians.png

To Convert Degrees in Radian with \(\pi \) Terms

Convert 12° to radians with \(\pi \) terms.

  1. Multiply 12° by \( \pi/180 \).
  2. Drop the °.
  3. Reduce.
  4. Simplify.
$$ 12^{\circ} \times \frac{\pi}{180} $$ $$=\frac{ 12^{\circ}\pi}{180} $$
$$=\frac{ 12\pi}{180} $$
$$=\frac{ \pi}{15} $$

Derivative

Trigonometry derivatives.png Radical derivatives.png

Integrals

$$ \int \sin x \: dx$$ $$= - \cos x + C$$ $$\int \arcsin x \: dx $$ $$= x \arcsin x + \sqrt{1-x^{2}} +C \:\:\:\: $$ $$\int \frac{1}{\sqrt{a^{2}-x^{2}}} \: dx $$ $$= \arcsin \frac{x}{a} + C $$
$$ \int \cos x \: dx$$ $$= \sin x $$ $$\int \arccos x \: dx $$ $$= x \arccos x - \sqrt{1-x^{2}} +C $$
$$ \int \tan x \: dx $$ $$= - \ln (\cos x) + C $$ $$ \int \arctan x \:dx $$ $$= x \arctan x - \frac{1}{2} \ln(1+x^{2})+ C \:\:\:\: $$ $$\int \frac{1}{a^{2}+x^{2}} \: dx $$ $$= \frac{1}{a} \arctan \frac{x}{a} +C$$
$$ \int \cot x \: dx$$ $$= \ln (\sin x) + C$$ $$ \int \text{arccot } x \: dx $$ $$= x \text{ arccot }x + \frac{1}{2} \ln(1+x^{2})+ C $$
$$ \int \sec x \: dx$$ $$= \ln ( \sec x + \tan x) + C$$ $$ \int \text{arcsec } x \: dx $$ $$= x \text{ arcsec } x - \ln \left[ x \left( 1+ \frac{x^{2}-1}{x^{2}} \right)\right]+C \:\:\:\: $$ $$\int \frac{1}{x\sqrt{x^{2}-a^{2}}} \:dx $$ $$= \frac{1}{a} \text{arcsec} \frac{x}{a} +C $$
$$ \int \csc x \: dx$$ $$= - \ln( \csc x + \cot x ) +C \:\:\:\: $$ $$\int \text{ arccsc } x \: dx $$ $$= x \text{ arccsc } x + \ln \left[ x \left( 1+ \frac{x^{2}-1}{x^{2}} \right)\right]+C $$
$$ \int \sec^{2} x \: dx$$ $$= \tan x +C $$
$$ \int \csc^{2} x \: dx$$ $$= -\cot x + C$$
$$\int x \cos x \:dx $$ $$= x \sin x + \cos x $$
$$\int x \sin x \:dx $$ $$= - x \cos x + \sin x $$
$$\int x \tan x \:dx $$ $$= \frac{1}{2} i \left( Li ( -e^{2ix}) + x^{2}+ 2ix \ln(1+e^{2ix})\right) $$

Identities

Trigonometry identities.png

Inverse Trigonometric Functions[2]

Inverse trigonometric functions are the opposite for the function.

Name Usual notation Definition Domain of x for real result Range of usual principal value
(radians)
Range of usual principal value
(degrees)
arcsine \(y = \arcsin(x), \: \sin^{-1}\) \(x = \sin(y) \) \(−1 \leqslant x \leqslant 1 \) \(−\frac{\pi}{2} \leqslant y \leqslant \frac{\pi}{2}\) \( −90° \leqslant y \leqslant 90° \)
arccosine \( y = \arccos(x), \: \cos^{-1} \) \( x = \cos(y)\) \(−1 \leqslant x \leqslant 1 \) \(0 \leqslant y \leqslant \pi \) \(0° \leqslant y \leqslant 180° \)
arctangent \(y = \arctan(x), \: \tan^{-1}\) \(x = \tan(y)\) all real numbers \(−\frac{\pi}{2} < y < \frac{\pi}{2} \) \(−90° < y < 90° \)
arccotangent \(y = \text{arccot}(x), \: \cot^{-1}\) \(x = \cot(y)\) all real numbers \( 0 < y < \pi\) \(0° < y < 180° \)
arcsecant \(y = \text{arcsec }(x), \: \sec^{-1}\) \(x = \sec(y) \) \( x \leqslant −1 \) or \( 1 \leqslant x \) \( 0 \leqslant y < \frac{\pi}{2} \) or \(\frac{\pi}{2} < y \leqslant \pi \) \(0° \leqslant y < 90°\) or \( 90° < y \leqslant 180°\)
arccosecant \( y = \text{arccsc}(x), \: \csc^{-1} \) \(x = \csc(y)\) \( x \leqslant −1 \) or \( 1 \leqslant x \) \(−\frac{\pi}{2} \leqslant y < 0 \) or \( 0 < y \leqslant \frac{\pi}{2} \) \(−90° \leqslant y < 0°\) or \( 0° < y \leqslant 90° \)

Triangle

Trigonometry triangle definition.gif

Trigonometry as a Hexagon

TrigIdentityHexagon.jpg

Internal Links

Parent Article: Mathematics