# Trigonometry/Functions

## Fundamental trigonometry functions

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

## Sine

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

## Cosine

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

## Tangent

• 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: ${\displaystyle \tan \theta ={\frac {opp}{adj}}}$

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

• "sine-opposite-hypotenuse
• cosine-adjacent-hypotenuse
• tangent-opposite-adjacent".

The remaining ratios are reciprocals of the previous ratios:

## Cotangent

• Cotangent θ is the reciprocal of tangent θ: ${\displaystyle \cot \theta ={\frac {adj}{opp}}}$

## Secant

• Secant θ is the reciprocal of cosine θ: ${\displaystyle \sec \theta ={\frac {hyp}{adj}}}$

## Cosecant

• Cosecant θ is the reciprocal of sine θ: ${\displaystyle \csc \theta ={\frac {hyp}{opp}}}$

## Other considerations

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

## Table

### Angle values

θ radians sinθ cosθ tanθ cotθ secθ cosecθ
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° 0 1 0 undefined 1 undefined

## Other resources

Basic Trigonometry (Youtube)
Basic Trigonometry II (Youtube)