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Laws of Sines & Cosines

The Hook

Solving the imperfect.

What is it?

Formulas to solve oblique (non-right) triangles.

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: GPS. Satellites use triangulation (technically trilateration) which relies on these laws to pinpoint your location on the globe.

How To Do It

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

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Booby Trap!

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

Real World Challenge

"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?"

The Logic: 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.

Training Drills

Law of Cosines

c² = a² + b² - 2abCos(C)

Looks like Pythagoras, with a correction factor for the angle.