Trigonometry Formulas

Determining Sides and Angles of a Right Triangle

Right triangle with its sides labeled h, a, and o and the interior angle opposite of side o labeled theta        a2 + o2 = h2 (Pythagorean theorem)
sin theta = o   csc theta = 1  =  h

 h  sin theta  o 
cos theta = a   sec theta = 1  =  h

 h  cos theta  a 
tan theta = o  =  sin theta        cot theta = 1  =  a

 a  cos theta tan theta  o 

Determining Sides and Angles of Any Triangle

Any triangle with the sides labeled a, b, and c and the opposite interior angles labeled alpha, beta, and gamma        sin alpha  =  sin beta  =  sin gamma    (Law of Sines)

a b c
c2 = a2 + b2 - 2ab cos gamma    (Law of Cosines)

Known Sides and Angles     Example
All three sides (SSS)
a, b, and c
cos alpha = b2 + c2 - a2     

Two sides and the angle
included between them (SAS)
b, alpha, and c
a2 = b2 + c2 - 2bc cos alpha
Two sides and an angle not
included between them (SSA)
a, b, and alpha
sin beta = b sin alpha     

One side and two angles (SAA)
aalpha, and beta
b = a sin beta     

sin alpha

Correcting Rotated Graph Axes

X and Y axes rotated about the origin by alpha degrees,  X' and Y' being the corrected axes.        alpha = tan-1(slope of x)

x' = x cos alpha + y sin alpha

y' = -x sin alpha + y cos alpha

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Page author: Dawn Rorvik (rorvikd@evergreen.edu)
Last modified: 03/28/1999