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Applications of Trigonometry

From solving triangles to modeling waves — trigonometry in the real world.

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.

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
The real power of trigonometry lies in its connections: it bridges geometry (shapes), algebra (equations), calculus (derivatives/integrals of trig functions), and differential equations (oscillatory solutions).