Laws of sines and cosines


Ratio of side to sine of subtending angle is same for all three ratios.
Equals 2R, the diameter of the circumcircle (circle that goes through the three vertices).

Can be used to solve triangle (i.e. triangulation) given any of these cases:


Generalization of Pythagorean Theorem.

Can be used to solve triangle (i.e. triangulation) given either of these cases:

Choose a case, enter the 3 values.

law of sines          law of cosines
AAS: A° A° S          SAS: S A° S
ASA: A° S  A°          SSS: S S  S
SSA: S  S  A°

   

Triangle: Perimeter=      Area=

α°   a      β°   b      γ°   c    

cos α= cos β= cos γ=
sin α= sin β= sin γ=
a/sin α= = b/sin β= = c/sin γ=

AAS: 90 45 1.414, 45 90 1, 60 60 1, 110 40 5
ASA: 90 1 45, 60 1 60
SSA: 5 4 90, 5 3.4202014 110, 1 1 60, 2 1 90, Bad: 1 3 10, 2 10 30 ambiguous: 1 2 30, 7 10 30

SAS: 1 90 1, 1 60 1
SSS: 1 1 1, 1 2 2, 3 4 5, 1 1 1.414