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Law of sines

Solving triangles

Japanese school year: Math I

What you learn

Learn the Law of Sines, which relates the sides and opposing angles of a triangle to its circumcircle radius. Building on basic trigonometric ratios and the inscribed angle theorem, it determines unknown lengths and angles. It is widely applied in surveying, navigation, and solving oblique triangles.

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Key points

Known as the Law of Sines, this links side lengths a,b,ca, b, c, opposite angles A,B,CA, B, C, and circumradius RR. It finds unknown sides, angles, or the circumdiameter 2R2R.

asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Derived from the Law of Sines, this formula calculates side length aa from circumradius RR and opposite angle AA. It also helps replace side lengths with sines in algebraic derivations.

a=2Rsin⁡Aa = 2R\sin A

This rule states that the ratio of side lengths a:b:ca:b:c is identical to the ratio of their opposite angles' sines sin⁡A:sin⁡B:sin⁡C\sin A : \sin B : \sin C. Use it to find side proportions from angles.

a:b:c=sin⁡A:sin⁡B:sin⁡Ca : b : c = \sin A : \sin B : \sin C

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