In This Lesson Law of Sines Law of Cosines 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 Waves & Oscillations Navigation & Surveying 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 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 AC Circuits: Voltage V(t) = V₀ sin(2πft)Mf1tJh7QkJZtvd9YUZnESE8lxDlA1X3cl/A7EVUeWjCzrYcnvWkBVDwbUCknUjSRWt1/o0GxPQHMaKWpdnwQMcTxJDDkSUlGvBCuApb/Mbxj9evAzfeCysyu66tcchDN4HJ3arfzvaZXLc0AyHyEb4cc4K8eONZACJ3GCcdaIPDJgPgVpz3xVl2BpWAOMtXpUfL7e+g9OfxNj+epdg29cL7cvL+2RfFdDH8TqZeNLm56uAsVRbIVi4lWh8y883lUSB6NJ6NKJtcP7g6PH9N8TpBoGij9VdyuittRerhREN/I0DJEc0tMcDsQUpVi1KsycbVXpeVbhfBPbOWlA3Epu82xtX7vHdpK7qIwY09vMNNz2WNOcGwyU83N/CJvXpB71WQvB/brUT55IuPF+5h77FwQvPZl7q1ZTq54QHeNw+kAube5RWUa2ZCfna/gULSvXjg32ZMGgPyJysE1wdevvSVzJ4FLLIWb349HTsIhrdkYkkAwWb5khNlEicEnmQIww+CHaBERan1dx739FvlFImKf/jOE2rL8QRhjOUCVxv/R43CL4sKBvcyutolHx9mwzv1LtXoa3eIJfvaC3KDl/NaP6eZk4usEvBlX1e6pNiK0ia4XQRd8XG7jfEw72vNsx9GbXjXyg8URjB1kloKXIh3vjHCaCO4zICyfsnPAH4BJ4rsvTLGmZlg33IAVdbeN/0grI9HdRW+yHe Quantum mechanics: Wave functions in Hilbert space 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 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 Surveying: Measuring distances using angles and the law of 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 Computer graphics: Rotation matrices use sin and cos Physics: Projectile motion, force decomposition along perpendicular axes