Mathematics.

oblique triangles

Law of Sines

Trigonometry20 minDifficulty4 out of 10

You should know: trigonometric functions

Overview

The law of sines relates the lengths of the sides of any triangle to the sines of its opposite angles. For a triangle with sides a, b, c opposite angles α, β, γ, the ratio of each side to the sine of its opposite angle is constant and equals the diameter of the triangle's circumcircle. The law lets you solve oblique (non-right) triangles by triangulation: given two angles and a side (AAS/ASA), or two sides and a non-included angle (SSA), you can find the remaining parts. The SSA case is the notorious 'ambiguous case', where the given data can correspond to zero, one, or two valid triangles.

Formal Definition

Definition

For a triangle with sides a, b, c opposite angles α, β, γ respectively, and circumradius R:

asin⁡α=bsin⁡β=csin⁡γ=2R\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} = 2R

Each side divided by the sine of its opposite angle equals the diameter of the circumscribed circle

Law of sines
sin⁡αa=sin⁡βb=sin⁡γc\frac{\sin\alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin\gamma}{c}

Equivalent reciprocal form

Worked Examples

  1. Apply the law of sines using the known side-angle pair (c, γ) and the unknown-angle side a.

    asin⁡α=csin⁡γ  ⇒  sin⁡α20=sin⁡40∘24\frac{a}{\sin\alpha} = \frac{c}{\sin\gamma} \;\Rightarrow\; \frac{\sin\alpha}{20}=\frac{\sin 40^\circ}{24}

  2. Solve for sin α, then take the inverse sine.

    α=arcsin⁡ ⁣(20sin⁡40∘24)≈arcsin⁡(0.5357)≈32.39∘\alpha = \arcsin\!\left(\frac{20\sin 40^\circ}{24}\right) \approx \arcsin(0.5357) \approx 32.39^\circ

  3. Note the ambiguous case: since a < c, there is only one valid triangle here (the supplementary angle 180° − 32.39° = 147.61° would make α + γ > 180°, so it is rejected).

    α≈32.39∘\alpha \approx 32.39^\circ

Answer: α ≈ 32.39°

Practice Problems

Difficulty 4/10

A triangle has angles α = 50°, β = 70°, and side a = 10. Find side b.

Difficulty 5/10

Surveyors sight a distant tower from two points 100 m apart on a baseline. The angles from the baseline to the tower are 65° and 78°. Find the distance from the first point to the tower.

Quiz

The law of sines is most directly used when you know:
In surveying, the law of sines enables triangulation, which finds:

Summary

  • Law of sines: a/sin α = b/sin β = c/sin γ = 2R, where R is the circumradius.
  • Used to solve triangles given AAS, ASA, or SSA data.
  • The SSA case is ambiguous: the given data may yield zero, one, or two valid triangles, since arcsin has two solutions in [0°, 180°].
  • Always check that computed angles sum to 180° with the given angle to rule out invalid solutions.

References

  1. WebsiteWikipedia — Law of sines