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Law of Sines
sin A
a
sin B
b
Law of Cosines (SSS form)
sin C
c
cos A
Law of Cosines (SAS form)
cos B
a
2
b
2
c
2
2bccos A
2accos B
b2
a2
c2
2
2
2
c
a
b
2abcos C
B
b
a
c2 a 2
2bc
a2
c2 b2
2ac
a2
b2 c2
2ab
These formulas refer to any triangle labeled with angles A, B and
C, and sides a, b and c. Each angle (always capitalized) must be
across from the side with the same letter (always lower case).
A
c
cos C
b2
C
The steps are named by which sides or angles are given in the
problem. Thus, if the measures of two sides and one angle are
given, and the angle connects the two sides, it is Side-Angle-Side
(written SAS). If the angle does not connect the sides, it is SSA
(Side-Side-Angle). This is the same notation from Geometry.
AAS Subtract to find third angle, then use Law of Sines to find the sides.
ASA Subtract to find third angle, then use Law of Sines to find the sides.
SAS Use Law of Cosines (SAS form) to find the unknown side (write it out to at least 6
decimal places), then use Law of Sines to find the smallest angle. Subtract to find the
third.
SSS
Use Law of Cosines (SSS form) twice to find two angles. Subtract to find the third.
SSA Draw two triangles with the same given information. Use Law of Sines to find an angle.
Put that angle into one triangle and the supplement of that angle in the other triangle (in
the same position). Cross out impossible triangles (where two angles add up to 180° or
more) and subtract to find the last angle of each possible triangle. Use Law of Sines to
find the last side for each possible triangle.