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. 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 C = 180° − 40° − 60° = 80°
b/sin 60° = 10/sin 40° → b = 10 sin 60°/sin 40° ≈ 13.47
Law of Cosines c² = a² + b² − 2ab·cos C
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 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 Sinusoidal functions model periodic phenomena throughout science:
y(t) = A sin(ωt + φ)
Sound: Musical notes as sums of harmonics → Fourier series Light: Electromagnetic waves are sinusoidal in E and B fields AC Circuits: Voltage V(t) = V₀ sin(2πft) Quantum mechanics: Wave functions in Hilbert space 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 The simple harmonic oscillator y'' + ω²y = 0 has solution y = A sin(ωt) + B cos(ωt) — connecting trig to differential equations .
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