The Law of Sines
Simple Explanation
In ANY triangle (not just right triangles), the ratio of a side's length to the sine of the angle opposite it is the same for all three sides: a/sinA = b/sinB = c/sinC.
Why Do We Need It?
This is the key tool for solving triangles that aren't right triangles, whenever you know two angles and a side, or two sides and a non-included angle.
See It
A scalene triangle with vertices A, B, C, and sides labelled a (opposite A), b (opposite B), and c (opposite C)
Formula
The Law of Sines
a/sin A = b/sin B = c/sin C
In any triangle, the ratio of a side's length to the sine of its opposite angle is the same for all three sides.
- a, b, c
- — the side lengths of the triangle
- A, B, C
- — the angles opposite sides a, b, c respectively
When to use it: Whenever you know two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA), of any triangle (not just right triangles).
Worked Example
Find a side using the Law of Sines
In triangle DEF, angle D = 30°, angle E = 90°, and side d = 5. Find side e.
Why Does This Work?
Dropping a perpendicular height h from one vertex to the opposite side lets that height be written two different ways using two different angles (e.g. h = b·sinA and h = a·sinB from the same triangle) — setting these equal and rearranging gives a/sinA = b/sinB, and the same argument extended to the third side gives the full three-way equality.
Real-Life Example
Triangulating a distant object in surveying
A surveyor measures the angles to a distant landmark from each end of a known baseline, but cannot directly measure the distance to the landmark.
The Law of Sines computes the unknown distance directly from the known baseline length and the two measured angles — the classic surveying technique of triangulation.
Practice
In triangle GHI, angle G = 45°, angle H = 45°, and side g = 8. Find side h.
HardCommon mistake
Mismatching sides and angles (e.g. pairing side a with sin B instead of sin A) — always pair each side with the sine of the angle directly OPPOSITE it.
Quick Review
- a/sinA = b/sinB = c/sinC — each side paired with the sine of its opposite angle.
- Use when you know two angles and a side (AAS/ASA), or two sides and a non-included angle (SSA).
- Derived by expressing the same height two different ways.