Law of Sines

Lesson 8 of 11 · 7 min · Free, no account needed

What you will learn

In any triangle, a ÷ sin A = b ÷ sin B = c ÷ sin C: each side over the sine of the angle across from it.
That common number is the diameter of the circle through the three corners.
Know two angles and one side, and you can find every other side: that is triangulation.

A lighthouse and two boats.
You know one distance and two angles. Can you find the rest? Bet first.
Drag any corner of the triangle.
Each side is divided by the sine of the angle across from it. Three goals.
What is that common number?
Draw the circle through the three corners and measure it.
This is the law of sines.
Name each side by the angle across from it: side a is across from angle A.
Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C.
All three equal the diameter of the circle through the corners. It works for every triangle, not just right ones.
Write the lighthouse equation.
Fill each box with the angle across from that side.
The angles add to 180°, so the lighthouse angle is 50°. AL is across from the 70° angle, the 5 km side is across from 50°: AL = 5 × sin 70° ÷ sin 50° ≈ 6.13 km.
A = 30°, a = 4 and B = 90°. How long is side b?
b ÷ sin 90° = 4 ÷ sin 30°, so b = 4 × 1 ÷ 0.5 = 8.
Game: rescue three boats.
Two coast stations 10 km apart see a boat. Work out its distance from S1, set the rope and Send.
Which side is the longest?
The angles are 40°, 65° and 75°. Tap the side.
Each side is the same number times the sine of its opposite angle. sin 75° ≈ 0.97 beats sin 65° ≈ 0.91 and sin 40° ≈ 0.64.
a ÷ sin A = b ÷ sin B = c ÷ sin C.
Back to the lighthouse, before and after. Drag it and the law measures boat A's distance.

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