= Trigonometry =

Some fundamentals of '''trigonometry''' are required for calculus.

<<TableOfContents>>

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== Unit Circle ==

The unit circle, showing major angles in radians and degrees, and showing the pair (cosine, sine):

{{attachment:unit.svg}}

Credit to [[https://commons.wikimedia.org/wiki/User:Crossover1370|Crossover1370]] on the other wiki, reproduced here under [[https://creativecommons.org/licenses/by-sa/4.0/deed.en|CC BY-SA 4.0 license]]

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== Functions ==

The '''trigonometric functions''' relate angles of a right triangle (or the unit circle: ''r = x^2^ + y^2^'') to the lengths of their sides.

||'''Function Name'''||'''Abbreviation'''||'''Ratio of'''        ||
||Sine               ||sin               ||opposite / hypotenuse ||
||Cosine             ||cos               ||adjacent / hypotenuse ||
||Tangent            ||tan               ||opposite / adjacent   ||
||Cosecant           ||csc               ||hypotenuse / opposite ||
||Secant             ||sec               ||hypotenuse /adjacent  ||
||Cotangent          ||cot               ||adjacent / opposite   ||

Then one of these functions is 'squared', as in ''sec^2^(x)'', this should be read as multiplying the function's output by itself. In other words, ''sec^2^(x) = sec(x) * sec(x)''.

The '''inverse trigonometric functions''' are arcsine, arccosine, and arctangent. The name derives from the fact that they are used to relate arc lengths to angles. There are variable notations:

 * arcsine of x = arcsin(x) = sin^-1^(x)
 * arccosine of x = arccos(x) = cos^-1^(x)
 * arctangent of x = arctan(x) = tan^-1^(x)

The '''hyperbolic trigonometric functions''' are parallels of these for the unit hyperbola: ''r = sqrt(x^2^ - y^2^)''.

||'''Function Name''' ||'''Abbreviation'''||'''Ratio of'''||
||Hyperbolic sine     ||sinh              || ||
||Hyperbolic cosine   ||cosh              || ||
||Hyperbolic tangent  ||tanh              || ||
||Hyperbolic cosecant ||csch              || ||
||Hyperbolic secant   ||sech              || ||
||Hyperbolic cotangent||coth              || ||

The '''inverse hyperbolic trigonometric functions''' are used to relate hyperbolic values back to a hyperbolic angles. There are variable notations. In particular, note that there is disagreement relating to ''ar-'' and ''arc-'' prefixes. Some notations prefer the latter to parallel the non-hyperbolic functions. On the other hand, as these functions do ''not'' relate to actual arcs but rather to area, other notations omit the ''c''.

 * inverse hyperbolic sine of x = arsinh(x) = sinh^-1^(x)
 * inverse hyperbolic cosine of x = arcosh(x) = cosh^-1^(x)
 * inverse hyperbolic tangent of x = artanh(x) = tanh^-1^(x)
 * inverse hyperbolic cosecant of x = arcsch(x) = csch^-1^(x)
 * inverse hyperbolic secant of x = arsech(x) = sech^-1^(x)
 * inverse hyperbolic cotangent of x = arcoth(x) = coth^-1^(x)



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