Solved Side c / Angle C
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Law of Sines Calculator
Solve a triangle from two angles and one side.
Comprehensive Guide
The Law of Sines links the sides of a triangle to the sines of their opposite angles. It is most useful when you know two angles and one side, because the missing parts of the triangle can then be found by proportional reasoning. That makes it a classic geometry tool for classwork, surveying, navigation, and construction layout.
Because the law ties each side to a matching angle, changing one known angle changes every other solved value. If one angle grows, the remaining angle shrinks, which then changes the side lengths through the sine relationship. The calculator helps you avoid repetitive trig work and keeps the triangle consistent.
Real-World Examples
A surveyor might measure one side of a triangle-shaped plot and two interior angles to find the remaining distances. Once one side is known, the law of sines can scale the rest of the triangle without needing direct access to every edge. That is useful when terrain makes some measurements difficult.
In navigation, a triangle made from landmarks or bearings can be solved if one distance and two angles are known. If the angle A changes from 30° to 35°, the resulting side lengths also shift. That sensitivity is exactly why the law is valuable: it transforms angle changes into real distance changes.
Reference Table
| Given | Solved | Use |
|---|---|---|
| A=30°, B=60°, a=10 | C=90°, b≈17.32, c=20 | Right triangle example |
| A=45°, C=55°, a=8 | B=80° | General triangle solve |
| A=20°, B=40°, a=15 | C=120° | Supplementary third angle |
Frequently Asked Questions
Do I need two angles and one side?
That is the most common and stable case. Once you know two angles, the third angle is fixed because all triangle angles add to 180°. After that, the side ratios fall into place through the sine rule.
Can there be more than one solution?
Yes, in some SSA situations the law of sines can produce an ambiguous case with two possible triangles. That is why triangle solvers need to be careful about what is given and whether the geometry is unique. A calculator should warn you when the setup is ambiguous.
Why use the law of sines instead of the law of cosines?
Use the law of sines when you already have angle-side opposite information or when two angles are known. The law of cosines is usually better when you know two sides and the included angle. They solve different triangle setups.
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