Law of Sines
Relates sides to opposite angles in any triangle.
Formula
Variables
Example
If a = 7, A = 30°, B = 45°: b = 7·sin45°/sin30° ≈ 9.9.
How it works
The Law of Sines connects each side of a triangle to the sine of the angle opposite it, so larger opposite angles correspond to longer opposite sides in a fixed proportion. It comes from comparing the areas of the same triangle written in different ways, since area can be expressed using two sides and the included angle. This makes the ratio side/sine(opposite angle) the same for all three pairs in any triangle.
When to use it
Use it when you know at least one side-angle opposite pair in a non-right triangle, especially for ASA, AAS, or sometimes SSA cases. It is common in surveying, navigation, and any geometry problem where you need to find an unknown side or angle in an oblique triangle.
Worked example, step by step
Given triangle ABC with a = 12, A = 50°, and B = 65°. Step 1: Find angle C. C = 180° - 50° - 65° = 65° Step 2: Use the Law of Sines to find b. b / sin 65° = 12 / sin 50° Step 3: Solve for b. b = 12 · sin 65° / sin 50° Step 4: Calculate. b ≈ 12 · 0.9063 / 0.7660 ≈ 14.2 Final answer: b ≈ 14.2 units
Common mistakes to avoid
- Pairing a side with the wrong opposite angle
- Using cosine instead of sine in the ratio
- Forgetting that the three angles in a triangle must add to 180°
Did You Know?
The law of sines lets surveyors and astronomers find distances to places they can never physically reach.
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