In This Lesson Law of Sines Law of Cosines Waves & Oscillations Navigation & Surveying mh7F5PyoEfeEIfA8l3IQ7k4BQZ7A7loxWmG2I3450L8zFHzXvKpVoiVMb/i141Y2K/Ms8yo850jxE5Fz02h2mU9REFmCIpZpHxGXoKSxWD6gYYP7IrNF/GuhIUOWz/5ppzGcNrfZ/wTC2GiBqI6Th2l//f6w6KBu0Z48hAD7/iRZ2tlQ2MgcR1+ibQ9NcnTb1UxIwOF9ZxSPRYP48mYf2bS629qxUHtER4bjosVywJqy8tdUCmmF5WbKQMI0ixoZekICRrpsLXT20qk3jbj02Aw0o1sgbaVeDoD690hdsECWYfQFl89DH96PZ4VHuHfhFgZhizRAks7ull7YWdGfH1RHINElDGXcWE6m6NFhGAhNpYX7PCSlOmZ/4WAsoU3jm5sQcH1XwEKqpEod9cMnTEh24IFRB6vk+3KVB3a6BJ3YM7Y3D+WAPFC2xuDzyb/82WC3+8MlmEu3umta5lO7FRABdJSQ0mXZhQ9LsLc4l8lBthwD3saPe0etcP7y5QMySOj5+MkI3gdU7q4jdgBfECW9NsUzdqadAHQt2cGZflS5f7y+lh7p4aCjkqsISbjouOZER3wanht2UXoB9U+yJeWFckeYJk+s7xW9ZaYcnuu9B2EmiC9KVRETqaSGi/bt4pG0xZalem9QN2XnXKN/0HAg458TAV3uRHxnmca8076BrjUh5zQcVfW+DnjtKPFXGratjtjOYE6I Law of Sines a/sin A = b/sin B = c/sin C = 2R (where R = circumradius)
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b/sin 60° = 10/sin 40° → b = 10 sin 60°/sin 40° ≈ 13.47
Law of Cosines 4eOiF+hO9UYoOFrxqG2E6gvFmnOqnYaX+MAlJWrUjK/kE6Bgg527QGcGTb9P3D6fJTdq2slvDYkEhWOvXUedkBluk/VA3wUuVEuO5JT0UsBliCd4bAzOKdAvWjebkD4PEErm9+IOUeb0peQoohVwu2t7I4vZ7Er4rz8TSRkVR9dvBiYH//PEwahhpBvobnGdqKhDy0YJCPj0sjMkhMtKSFDU1b69WkQMowClDpPlFVXrEBYDSq715mxAAtl9RiNvfMFgBwBG8uc4c/ujkxQmUeln/ZPiGeAtD9STI56JdnupSAsTEWfV8awYvwYWoIAeQ0vqsN3Sj9Y5iqiA5eZ3d/VN4xKQAuxd2Xa6lHnKrEUieOj0xDbvSp1iMY/qJyaxcHjGSZSitQimuGupIHshm75uWWj1wEqZWCX8BKeLWX9VVuNPLoKxJwiuqU9nVSDfJN2yEMrkGjFGJh4A0XtdkqauQP3hSRqWUXml3W65KsPsmPSYpPgpT7hgLWL2+8a0uzmjhLD6YFkF8LLr0xDynHaEJuOpGxk7/WCFx01iEkT23ocPeJZcQ1agxQBULO/0wvzyarPB7CvAqB2Urm+lw/u6HuzGkWYVqNmbfDC7J4TfUJOdnlXxg/k761VumV+XEK0G8imxwy4ZX1UbBAN8iHdL4+vI5vXjje1oT4sxLDPWA0IPRouU1EtDrpLre3Kw9u/x9vRVohQZ 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 The simple harmonic oscillator y'' + ω²y = 0 has solution y = A sin(ωt) + B cos(ωt) — connecting trig to differential equations .
Navigation & Surveying 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 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 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