# 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 $\cos \theta <1\,$ 