Trigonometry/Functions

Fundamental trigonometry functions

There are six trigonometric functions in Trigonometry: sine, cosine, tangent, cotangent, secant, and cosecant.

Sine θ is the length of the leg opposite θ over the length of the hypotenuse: $\sin \theta ={\frac {opp}{hyp}}$

Cosine θ is the length of the leg adjacent to θ over the hypotenuse: $\cos \theta ={\frac {adj}{hyp}}$

Tangent of θ is the length of the leg on the opposite side of the triangle from the angle θ over the length of the leg of the triangle adjacent to the angle θ: $\tan \theta ={\frac {opp}{adj}}$

These three can be memorized by use of the name of the princess "Soh Cah Toa," meaning:

The remaining ratios are reciprocals of the previous ratios:

Since the hypotenuse of a right triangle is always the longest side, $opp<hyp\,$ and $adj<hyp\,$
If we divide both sides of each of these inequalities by the positive number $hyp\,$ , we get ${\frac {opp}{hyp}}<{\frac {hyp}{hyp}}\,$ and ${\frac {adj}{hyp}}<{\frac {hyp}{hyp}}\,$
or $\sin \theta \leq 1\,$ and $\cos \theta \leq 1\,$

θ	radians	sinθ	cosθ	tanθ	cotθ	secθ	cosecθ
0°	0	0	1	0	undefined	1	undefined
30°	π/6	1/2					2
45°	π/4			1	1
60°	π/3		1/2			2
90°	π/2	1	0	undefined	0	undefined	1
180°	π	0	-1	0	undefined	-1	undefined
270°	3π/2	-1	0	undefined	0	undefined	-1
360°	2π	0	1	0	undefined	1	undefined