# 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: $\sin \theta ={\frac {opp}{hyp}}$

## Cosine

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

## 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: $\tan \theta ={\frac {opp}{adj}}$

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

• "sine-opposite-hypotenuse

The remaining ratios are reciprocals of the previous ratios:

## Cotangent

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

## Secant

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

## Cosecant

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

## Other considerations

• Since the hypotenuse of a right triangle is always the longest side, $opp  and $adj
• 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 $\sin \theta <1\,$  and $\cos \theta <1\,$