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Laws of Sines & Cosines

Overview

Solving the imperfect.

What is it?

Formulas to solve oblique (non-right) triangles.

History

Vital for the Great Trigonometrical Survey of India, which mapped the height of Mt. Everest.

Key Idea

Law of Sines for pairs (Angle A + Side a). Law of Cosines for SAS or SSS.

Practice This Topic

Concept Guide

Plain English: SOH CAH TOA fails if there isn't a 90-degree angle. These Laws fix that. Law of Sines says the ratio of 'Big Mouth (angle) eats Big Meal (side)' is always constant.

Real-world example: GPS. Satellites use triangulation (technically trilateration) which relies on these laws to pinpoint your location on the globe.

How to do it

  1. Check your triangle info.
  2. Do you have a 'matching pair' (an angle and its opposite side)? Use Law of Sines.
  3. Do you have Side-Angle-Side (SAS) or Side-Side-Side (SSS)? Use Law of Cosines.
  4. Plug and play the variables.
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Common Pitfall

The Ambiguous Case. In Law of Sines, sometimes one set of numbers can create two different possible triangles.

Word Problem
"Two rangers are at towers 10 miles apart. Tower A sees a fire at 40 degrees. Tower B sees the fire at 60 degrees. How far is the fire from Tower A?"
Reasoning: We know 2 angles (40, 60), so the 3rd is 80 (180-40-60). We have a pair: Angle 80 matches Side 10. Use Law of Sines: 10/Sin(80) = x/Sin(60). Solve for x.

Practice Examples

Law of Cosines
c² = a² + b² - 2abCos(C)
Looks like Pythagoras, with a correction factor for the angle.