Law of Cosines & Law of Sines w/Area

K (Law of Cosines)

Law of Cosines, Heron's Formula

U (Law of Cosines)

Trigonometry can be applied to triangles that are not right (oblique).

D (Law of Cosines)

Use the Law of Cosines to solve SSS and SAS triangle scenarios.

EQ (Law of Cosines)

How do I solve oblique triangles?

Law of Cosines – Standard Form

a² = b² + c² – 2bc·cos A; b² = a² + c² – 2ac·cos B; c² = a² + b² – 2ab·cos C

Law of Cosines – Alternative Form

cos A = (b² + c² – a²)/(2bc); cos B = (a² + c² – b²)/(2ac); cos C = (a² + b² – c²)/(2ab)

Heron's Formula (for area)

Area = √[s(s – a)(s – b)(s – c)], where s = (a + b + c)/2

K (Law of Sines)

Law of Sines

U (Law of Sines)

Trigonometry can be applied to triangles that are not right (oblique).

D (Law of Sines)

Use the Law of Sines to solve AAS, ASA, and SSA triangle scenarios.

EQ (Law of Sines)

How do I solve oblique triangles?

Law of Sines (Main Formula)

a/sin A = b/sin B = c/sin C

Ratio of sine values (check)

sin A / a ≈ sin B / b ≈ sin C / c (should be close if measured correctly)

Area of an Oblique Triangle

Area = (1/2)ab·sin C = (1/2)bc·sin A = (1/2)ac·sin B

SSA Ambiguous Case – 2 triangles

b > a > h → 2 triangles possible

SSA Ambiguous Case – 1 triangle (right)

a = h → exactly 1 triangle

SSA Ambiguous Case – 1 triangle (longer side)

a > b → 1 triangle

SSA Ambiguous Case – no triangle (shorter side)

a < b → 0 triangles

SSA Ambiguous Case – no triangle (less than height)

a ≤ b → 0 triangles