In This Lesson g8eeTZduqumJQ1VQVMvM3oY7GG2JJKPfonnFwVEIEAiGst/BHNfMwRsVOGb92nxty/7OaGwx9pug/L3fzJcyrOx2oanSD4lRBF8m8yu6+ORTyZWhdqcoaLzWUJ0GZmP397clNpq4l2uK0TwtDDb+Pjt0sToLQDk2nC0oG4WUSKo1eQqBMAud+ch69hNbL9rb8aFzug0c4uCuMb1DPm3kw4jFqhDCHdqwzIPlV7s3YvzWE/pVzmxexltfBv2u3clN+kmkrkRH4k11cMEFw3CmbXWATx8/J2HbOjWZa5U+QAnkvA7USAKCTn0WjgtW1lgxs//NH0BQxx+p/XjUncDxXIppsycljaUtni4qMqLYiLNwu5U316hIRcJyqKeaLpivdcIkv5AZLBaq0aGs59+iTcjreukv+8EbUYbEhbtNrizffQ/zcdDaYaQPN4p9scp4JndgricBt7lBHWxO2mdcOp+uDcCUNMlGH08jPZ3EPJH285E/ccYrFmwxOaWvSMbF6ZuGm4qczynKqLpZpvU7Mq31XdNXLP5TpOuHsHYK/6KYhtHJYYtWILUnjcepT+ZbEyeZSoOVB+nkLGoL3gw6yC8j2e1ZCtExJQ/JIKcsyt/RKXyxue3+GyHXdPd/LOjDn1srHHUGCY5R+KSvuVsiw/8l7+wAKhJ68bD90X5457iii18TTbmUk2G3MPyySpdKaHre Law of Sines Law of Cosines 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 Waves & Oscillations Navigation & Surveying BUmtNGDQd3buX3JJoFXIGgS5aqe0eSSN/1Wu3HPZwyPFs/bGIfa9E++6FcOOHFSDHxxVoCO5V6baavjUI6nok2RTgy6rvPKbsPL0FcWmDYXJrmQ2nGScbjq55Dxv9SHeXfhNEtwTldjtoSyu3/Ff3a8TgSzNvfv1tz3NhSK/4BfhoGtFHx09dfuZrvo9Zv8ZhhvfpmlVMTvK2vdR1rUCNTMT6IkEvTZBdGLz3uE3EvwyAFOtZC3YHk9CS4dhHhYWXAYLymV6nxDhnnFrMo4chmKUv58sUYFfOlCu4ongfrafsmZkCKfW4zqhOF7PqTSuEVd2hZwTxaPiIA0XNsC4In2x0x3d8uzsEWdWJnQtbx+FLPQ4oSi/3vpvhsJD+iKP4FPPWKi5roUL1Do/pUithrh3i81iIF+4tuWJ6XRmS2S/QmLQphgiXyKxCkBT4oKjeuFPMEyvX0Xd0sav5pnS/4lWAQOZKf//tY/Tl4jt182Y5B/ZGq9nxx8e7Myn/dpqdd7IBVkmwHVkzLyRwDdNKz7UIh7yuH+2YWx+DkMkwm+P4Lzc1sRDELcHMxNOLjp7PchYwffapMw5T5o4WDCZztP8tiSN8FGDbZ3YckZpDAjMDWo1vMzWB/n1nZIOBUJ2sw2SyhCWki6QpaFckbxEj5MHQDsTNZ/fPkCDRNm8EqRkwz0pxZnR2tkFMHG6CLu7LOeh 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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
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 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)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 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 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 xBZqdZD2quD3xeOW7bicnHBLzDfNWWec6xw1qD8vIlAOyLfgSW2TPSaW1YBA7Bn09P4BMddT3LHTc0f3K7d/3ROho4vIlrWFq69Rakf/bCptyM/86nmUkFNlhNwBOBvtZe4aPcHZ1RMI0LHw73kHXiejCU4bAoo+Y6i1/3cLxoOUQuxKqVxUyY84VC55lpBxrZYV4+mVDGShl7r/ppWdGS6v9figNO5IAn23Pd3wPjqfUksv6KsmkvgBiKHGMRlyTdhB/nYsNkakXFJMdET+tpPwhDQ6UzUjWQFBKAfT60RJOKetwMQR1t7qvVlCcOmM/7QI5ddy5Dz8IOWQoLhzBk8XKaLnU29Z/bPFuDuR2r6x05nWAGu9cTGT9zmZVeD4qsQKLMSvfUW/iGd6a0w+l0mP7JrSyI2wstKEelUgsqDrb1rAKxqXe+ZYSrQwLcI9gVnd1gzp7yknj6RWy34e09OWXxSjIbhONa5TkEDyYiXMztBrqdXs3s3rtE1lsMMWc0/IRB2unIEH6JudLLuec4C93YUG2cDu0pE3VkuXsWr34TWe3Ax+bK9KNaXlwGtob4ZyN+4cKKvEZoI3u8bUsKdokMffFr3E4XzW/YL9BWqa+ZSzof5BW23HHTZyyEwhEKRa1Daj67l8v4JpmZDkojLK3wiyFX8uImnoj4MBIXdJxeMO4LSGhik+YyB0UhvvLOXc