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.

File:TrigTriangle.svg

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 $\sin \theta \le 1$ and $\cos \theta \le 1$

Table

Angle values

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

Quiz

• Trigonometry/Functions/Quiz

Other resources

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

Basic Trigonometry (Youtube)

Basic Trigonometry II (Youtube)

• Trigonometry/Functions/Flash cards