Law of Sines
Relates sides of any triangle to the sines of opposite angles.
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 with the sine of the angle directly opposite it, so the ratio stays the same for all three side-angle pairs. It comes from comparing the areas of the same triangle written in different ways, which reveals a shared constant tied to the triangle’s circumradius. This makes it especially useful when you know a side and its opposite angle, because it lets you solve for another side or angle.
When to use it
Use this when you are working with a non-right triangle and you know at least one side-angle opposite pair. It is common in surveying, navigation, and geometry problems where triangles are formed by distances and bearings.
Worked example, step by step
Given a triangle with A = 35°, B = 60°, and a = 12 Step 1: Use the Law of Sines: b/sin B = a/sin A Step 2: Solve for b: b = 12·sin60°/sin35° Step 3: Substitute values: b = 12·0.8660/0.5736 Step 4: Calculate: b ≈ 18.1 Final answer: b ≈ 18.1 units
Common mistakes to avoid
- Using the formula with only side lengths and no opposite angle information
- Pairing a side with the wrong opposite angle
- Forgetting to use degrees in the sine function when the angle is given in degrees
Did You Know?
The law of sines lets surveyors and astronomers find distances to places they can never physically reach.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.