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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.

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

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).

Related Topics

Triangles & ProofsThe triangle geometry that trig applications depend on. Second-Order DEsOscillatory solutions — where trig meets differential equations. Vectors & MatricesDot products, projections, and the law of cosines generalized. IntegralsTrigonometric substitution and integration of trig functions.