Applications Summary

Trigonometry · Axiom Academy

SUMMARY Applications of Trigonometry Let's review how trigonometry extends beyond right triangles to solve any triangle and model real-world phenomena from engineering to navigation. The Formula: For any triangle with sides a, b, c opposite angles A, B, C : a/sin(A) = b/sin(B) = c/sin(C) When to Use: Apply Law of Sines for AAS (two angles, one side) and ASA (angle-side-angle) configurations The Ambiguous Case (SSA): Two sides and an angle opposite one of them can yield 0, 1, or 2 possible triangles—check all solutions carefully Key Insight: The ratio of any side to the sine of its opposite angle remains constant throughout the triangle The Formulas: c² = a² + b² - 2ab·cos(C) and cyclic variations for other sides When to Use: Apply Law of Cosines for SAS (two sides, included angle) and SSS (all three sides known) configurations Generalization: Law of Cosines reduces to Pythagorean theorem when C = 90° (since cos(90°) = 0) Finding Angles from Sides: Rearrange to cos(C) = (a² + b² - c²)/(2ab) when all three sides are known SAS Formula: When two sides and included angle are known: Area = (1/2)ab·sin(C) Heron's Formula (SSS): When all three sides are known: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semi-perimeter Application: Area formulas eliminate the need to find a base and height—use whatever information is available Surveying & Engineering: Calculate irregular plot areas by triangulation, breaking complex shapes into triangles

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