The Law of Sines: Every Triangle Is a Circle Trying to Express Itself

The Law of Sines says:

a/sin A = b/sin B = c/sin C

This looks like a formula for triangles. But here's the unlock: it's actually telling you that every triangle sits inside a unique circle, and the sides are related to that circle's diameter.

The ratio a/sin A equals the diameter of the circle that passes through all three vertices. Every triangle is inscribed in exactly one circle. The Law of Sines is that circle talking.

This isn't a triangle formula. It's a circle formula wearing a triangle costume.

What the Law Says

For any triangle with:

• Sides a, b, c
• Opposite angles A, B, C (angle A is opposite side a, etc.)

The ratios of side to sine of opposite angle are all equal:

a/sin A = b/sin B = c/sin C = 2R

where R is the radius of the circumscribed circle (the circle passing through all three vertices).

The Circle Connection

Every triangle has a circumcircle — a unique circle that passes through all three vertices.

Here's why the Law of Sines involves this circle:

Take angle A. The arc opposite to A (the arc from B to C not passing through A) is related to angle A by the inscribed angle theorem: the inscribed angle is half the central angle.

When you work through the geometry, you find that: a = 2R sin A

Rearranging: a/sin A = 2R

The same is true for the other sides. They all equal 2R, the diameter.

When to Use the Law of Sines

Use it when you have:

• Two angles and one side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA)

Example 1: In a triangle, angle A = 40°, angle B = 60°, and side a = 10. Find side b.

First, note that C = 180° - 40° - 60° = 80°.

Using Law of Sines: a/sin A = b/sin B 10/sin 40° = b/sin 60° b = 10 × sin 60° / sin 40° b = 10 × 0.866 / 0.643 ≈ 13.5

Finding Angles

You can also use it to find angles:

Example 2: In a triangle, a = 8, b = 12, and A = 30°. Find angle B.

a/sin A = b/sin B 8/sin 30° = 12/sin B sin B = 12 × sin 30° / 8 = 12 × 0.5 / 8 = 0.75 B = arcsin(0.75) ≈ 48.6°

The Ambiguous Case (SSA)

When you have two sides and an angle opposite one of them, sometimes there are two possible triangles.

If a = 8, b = 12, and A = 30°, we found sin B = 0.75, so B ≈ 48.6°.

But sin(180° - 48.6°) = sin(131.4°) = 0.75 too!

Both B = 48.6° and B = 131.4° are mathematically valid. You need to check which one (or both!) makes a valid triangle.

If A + B must be less than 180°:

• A + 48.6° = 78.6° < 180° ✓ (valid)
• A + 131.4° = 161.4° < 180° ✓ (also valid!)

So there are two different triangles that satisfy these conditions.

Why the Ambiguity Happens

Geometrically: if you have side a, angle A, and side b, you're trying to "swing" side b to meet side a's endpoint.

Sometimes b is long enough to hit in two places. Sometimes it only reaches one place. Sometimes it's too short to reach at all.

The ambiguous case only happens when you're given SSA with the angle not between the two sides.

The Area Formula

The Law of Sines leads to an area formula:

Area = (1/2) × a × b × sin C

This uses two sides and the included angle.

Why? Area = (1/2) × base × height. If the base is a and the height is b × sin C (the perpendicular height from the angle), you get this formula.

Comparison to Law of Cosines

Law of Sines is good when you have:

• Angle-angle-side (AAS)
• Angle-side-angle (ASA)
• Side-side-angle (SSA, with ambiguity warning)

Law of Cosines is good when you have:

• Side-side-side (SSS)
• Side-angle-side (SAS)

Both laws can solve any triangle. The question is which one is more direct for your given information.

Proof Sketch

Drop an altitude h from vertex C to side c.

In the left triangle: sin A = h/b, so h = b sin A In the right triangle: sin B = h/a, so h = a sin B

Since h = h: b sin A = a sin B a/sin A = b/sin B

The same reasoning with other altitudes gives the full law.

Applications

Surveying: Measuring distances to inaccessible points by measuring angles from known positions.

Navigation: Determining position from angular bearings.

Astronomy: Calculating distances to stars using parallax angles.

Whenever you can measure angles more easily than distances, the Law of Sines helps.

The Core Insight

The Law of Sines connects every triangle to its circumscribed circle.

The ratio a/sin A isn't arbitrary — it's the diameter of the unique circle passing through all three vertices. Bigger triangle, bigger circle, bigger ratio.

This is why the law works: it's not really about triangles. It's about the relationship between inscribed angles and the arcs they subtend.

Every triangle is a circle trying to express itself through three points. The Law of Sines is that expression made algebraic.

Part 6 of the Trigonometry series.

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