The laws of sines and cosines

Use the laws of sines and cosines with ambiguity checks

Problem types

Sine, cosine, and tangent
Deriving the laws of sines and cosines · Useful idea
The altitude derivation of the laws of sines and cosines shows which measurements each law needs. Solving an oblique triangle applies the matching law to those measurements.
Solving trigonometric equations · Useful idea
Finding an angle of a triangle from the law of sines solves an equation in the sine, which can have two solutions.
Velocity and force as vectors · Useful idea
Two forces or velocities that are not perpendicular form a triangle with their resultant, so the law of cosines gives the resultant's magnitude and the law of sines its direction.
When given measures make a triangle · Useful idea
Three sides can form a triangle only if the triangle inequality holds, which a solver of oblique triangles checks first.

Vectors and their components · See it applied
The magnitude and direction of a sum of two vectors are found by solving the triangle they form with the law of cosines and sines.
The incircle and the circumcircle · See it applied
In any triangle the law of sines gives a over sin A equal to the diameter of the circumscribed circle.

Area from two sides and the included angle · Related idea
The height used to count triangles in the ambiguous case is b sin A, the same product that appears in the area formula one half ab sin C.
Sine and cosine of complementary angles · Shared foundations
Side ratios of special right triangles · Shared foundations

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