In This Lesson 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 Law of Sines Law of Cosines Waves & Oscillations Navigation & Surveying 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 Law of Sines 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°
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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 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 Physics: Projectile motion, force decomposition along perpendicular 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