Law of Sines

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

In any plane triangle the ratio of a side to the sine of its opposite angle is constant and equal to the circumdiameter 2R. Use with AAS, ASA, or SSA (the ambiguous case).

$a, b, c$ (m): Side lengths
$A, B, C$ (deg or rad): Opposite angles
$R$: Circumradius, a/(2 sin A)

Step-by-step solved example

A triangle has a = 8 m, A = 30°, B = 45°. Find side b.

Law of sines uses the same oblique triangle as the handbook figure.

1. Write the proportion
b / sin 45° = 8 / sin 30°.
$\frac{b}{sin 4 5 ^{\circ}} = \frac{8}{sin 3 0 ^{\circ}}$
2. Solve
sin 30° = 0.5, sin 45° = √2/2 ≈ 0.7071. b = 8(0.7071)/0.5 = 11.31 m.
$b = 11.31 m$

Answer: b = 11.31 m

10 practice questions

0/10 correct

1.a = 8, A = 30°, B = 90°. Side b?

2.The law of sines applies to

3.a = 10, A = 30°, C = 90°. Hypotenuse c?

4.If A = B then the law of sines implies

5.sin 90° in a/sin A = c/sin 90° equals

6.a = 6, b = 6, A = 70°. Angle B is

7.Circumradius R = a / (2 sin A). If a = 10 and A = 90°, R =

8.The SSA ambiguous case can occur when

9.For a 45-45-90 triangle with leg a = 1, a / sin A equals

10.When C = 90°, the law of sines reduces to