Trigonometry

The Original Three

Sine (sin)

For a right triangle, the sine of an angle is the ratio of the opposite side from the angle divided by the hypotenuse:

For both acute and obtuse triangles (all angles less than 90 and an angle greater than 90, respectively, the Law of Sines applies as follows:

For a unit circle (a circle with radius 1, centered at the origin), the sine is the y-coordinate of the point of the circle at a specified angle.

The Organic Chemistry Tour provides a fun tutorial into some real-life applications for trigonometry below:

Cosine (cos)

For a right triangle, the cosine of an angle is the ratio of adjacent side of the angle divided by the hypotenuse:

For both acute and obtuse triangles (all angles less than 90 and an angle greater than 90, respectively, the Law of Cosines applies as follows:

For a unit circle (a circle with radius 1, centered at the origin), the cosine is the x-coordinate of the point of the circle at a specified angle.

Tangent (tan)

The tangent of an angle is the opposite side divided by the adjacent side (not hypotenuse); also the sine divided by cosine where cosine does not equal 0:

For a unit circle, the tangent is y divided by x. Note: tangent is undefined when the cosine of the angle equals 0.

Inverses, Not Reciprocals!

Arcsine (arcsin, asin, sin^-1)

The arcsine is the inverse of the sine, thus the angle:

The Law of Sines can also come in handy here:

Arccosine (arccos, acos, cos^-1)

Arccosine is the inverse of the cosine, thus the angle:

The Law of Cosines can also come in handy here:

Arctangent (arctan, atan, tan^-1)

Arctangent is the inverse of the cosine and is defined as follows:

Now The Reciprocals

Cosecant (csc)

Since sine is:

the cosecant is the reciprocal of the sine, defined as follows:

Whereas sine goes up a triangle from the opposite, the cosecant goes down the triangle.

Note: the sine cannot equal zero!

Secant (sec)

Given cosine is:

the secant flips the fraction:

The cosine stays along the base while the secant follows "what goes up, must come down" (up to the hypotenuse, then back down to the base).

Note: the cosine cannot equal zero!

Cotangent (cot)

Given that tangent is (rise over run):

the cotangent flips the above on its head:

the steeper the tangent the smaller the cotangent, aka run over rise

Arccosecant (arccsc, acsc, csc^-1)

It is the inverse of the cosecant:

The cosecant doesn't exist at 0, therefore the arccosecant only works for:

Arcsecant (arcsec, asec, sec^-1)

This is the inverse of the secant:

Secant values don't exist between -1 and 1, therefore the range is:

Arccotangent (arccot, acot, cot^-1)

This is the inverse of the cotangent:

Cotangent values can be any real number, so the range is as follows:

The Hyperbolic Functions

Full definitions and applications of the following are available at Wolfram MathWorld.

Hyperbolic Sine (sinh)

The definition of the rollercoaster-loving hyperbolic sine is as follows:

Hyperbolic Cosine (cosh)

The definition of the hammock-shaped hyperbolic cosine is as follows:

Hyperbolic Tangent (tanh)

The definition of the elongated s-curve hyperbolic tangent is as follows:

Hyperbolic Arcsine (arcsinh, asinh, sinh^-1)

The definition of the hyperbolic arcsine is as follows:

where the range of x is:

Hyperbolic Arccosine (arccosh, acosh, cosh^-1)

The definition of the hyperbolic arccosine is as follows:

where the range of x is:

Hyperbolic Arctangent (arctanh, atanh, tanh^-1)

The definition of the hyperbolic arctangent is as follows:

where the range of x is:

Hyperbolic Cosecant (csch)

The definition of the hyperbolic cosecant is as follows:

where the range of x is:

Hyperbolic Secant (sech)

The definition of the hyperbolic secant is as follows:

where the range of x is:

Hyperbolic Cotangent (coth)

The definition of the hyperbolic cotangent is as follows:

where the range of x is:

Hyperbolic Arccosecant (arccsch, acsch, csch^-1)

The definition of the hyperbolic arccosecant is as follows:

where the range of x is:

Hyperbolic Arcsecant (arcsech, asech, sech^-1)

The definition of the hyperbolic Arcsecant is as follows:

where the range of x is:

Hyperbolic Arccotangent (arccoth, acoth, coth^-1)

The definition of the hyperbolic Arccotangent is as follows:

where the range of x is:

Another way to put it is: