Given a right triangle, select a vertex where the hypotenuse and one of the legs meet.

The angle #theta# created by this two sides will be used as a reference point

The side that formed the angle #theta# together with the hypotenuse will be referred to as #adjacent# (side adjacent to the angle). The other side will be referred to as #opposite# (side opposite the angle)

The ratio between the #opposite# and the #"hypotenuse"# is called #"sine" (sin#). The inverse of this ratio is called #"cosecant" (csc#)

#sin theta = "opposite" / "hypotenuse" #

#csc theta = "hypotenuse" / "opposite" = 1 / sin theta#

The ratio between the #adjacent# and the #"hypotenuse"# is called "cosine". The inverse of this ratio is called "secant"

#cos theta = "adjacent" / "hypotenuse" #

#sec theta = "hypotenuse" / "adjacent" = 1 / cos theta#

The ratio between the #opposite# and the #adjacent# is called

#"tangent"#. The inverse of this ratio is called #"cotangent"#

#tan theta = "opposite" / "adjacent"#

#cot theta = "adjacent" / "opposite" = 1 / tan theta#

For example, in a 30-60-90 triangle

#sin 30 = 1 / 2#

#cos 30 = 3^(1/2)/2#

#tan 30 = 1 / 3^(1/2)#

#csc 30 = 2#

#sec 30 = 2/3^(1/2)#

#cot 30 = 3^(1/2)#

#sin 60 = 3^(1/2)/2#

#cos 60 = 1 /2 #

#tan 60 = 3^(1/2)#

#csc 60 = 2/3^(1/2)#

#sec 60 = 2#

#cot 60 = 1 / 3^(1/2)#