Trigonometry/Functions

• See also Trigonometry/Polar for an approach that is useful for -∞<θ<∞

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

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, $opp<hyp\,$ 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 $\sin \theta <1\,$ and $\cos \theta <1\,$

Table

• See Table of functions.

Quiz

• Trigonometry/Functions/Quiz

Other resources

• Reading: w:Trigonometric_Functions (Wikipedia)
• Videos:

Basic Trigonometry (Youtube)

Basic Trigonometry II (Youtube)

• Trigonometry/Functions/Flash cards