In This Lesson Law of Sines Law of Cosines Waves & Oscillations Navigation & Surveying Law of Sines 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 a/sin A = b/sin B = c/sin C = 2R (where R = circumradius)
Used when you know AAS, ASA, or SSA (the ambiguous case — check for 0, 1, or 2 solutions). The connection to the circumscribed circle radius R is elegant geometry.
Example: A = 40°, B = 60°, a = 10. Find b. C = 180° − 40° − 60° = 80°
b/sin 60° = 10/sin 40° → b = 10 sin 60°/sin 40° ≈ 13.47
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This is the Pythagorean theorem generalized to all triangles . When C = 90°, cos C = 0 and it reduces to a² + b² = c². It also defines the dot product of vectors in linear algebra .
Waves & Oscillations Sinusoidal functions model periodic phenomena throughout science:
y(t) = A sin(ωt + φ)
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Navigation & Surveying Trigonometry was invented for navigation and astronomy. Modern applications include:
GPS: Triangulation using satellite signals Surveying: Measuring distances using angles and the law of sines Computer graphics: Rotation matrices use sin and cos Physics: Projectile motion, force decomposition along perpendicular axes