Lesson 8.3.3

Intro to Law of Sines

SOH CAH TOA only works for right triangles. The Law of Sines extends trigonometry to any triangle by relating each side to the sine of its opposite angle.

Introduction

Many real triangles — land plots, satellite paths, roof trusses — are not right triangles. The Law of Sines lets you solve these oblique triangles when you know AAS, ASA, or SSA (two angles and a side).

Past Knowledge

Sine (8.2.5). Inverse sine (8.3.1). Triangle angle sum = 180°.

Today's Goal

State, prove, and apply the Law of Sines to solve non-right triangles.

Future Success

Law of Cosines (8.3.4), ambiguous case (SSA), pre-calculus.

Key Concepts

Law of Sines

Equivalently:

When to Use It

AAS — two angles and a non-included side
ASA — two angles and the included side
SSA — two sides and an angle opposite one (⚠️ ambiguous case possible)

Theorem & Proof

Two-Column Proof: Law of Sines

Given: with sides opposite angles

Prove:

Strategy: Drop an altitude and express it two ways using sine.

#	Statement	Reason
1	Draw altitude from to	Every triangle has an altitude
2	In the right triangle on the left: →	Definition of sine in a right triangle
3	In the right triangle on the right: →	Definition of sine in a right triangle
4		Transitive property (both equal )
5		Divide both sides by

∎ Repeat with another altitude to include . The key insight: the same altitude can be expressed as sine-times-a-side from two different angles.

Worked Examples

Basic

AAS — Finding a Side

. Find .

Intermediate

ASA — Solving Completely

. Solve the triangle.

→

Advanced

SSA — Finding an Angle

. Find angle .

Common Pitfalls

The Ambiguous Case (SSA)

When given two sides and an angle opposite one of them, there may be 0, 1, or 2 solutions. If , no triangle exists. If and the given angle is acute, check if also works.

Not Pairing Correctly

Side pairs with angle , with , etc. Each side must go with its opposite angle.

Real-Life Applications

Surveying — Land Measurement

Surveyors can measure angles from two known points to a distant landmark, then use the Law of Sines to compute the distance to the landmark without physically traveling there.

Astronomy — Stellar Distance

The parallax method for measuring star distances creates a very thin triangle. The Law of Sines computes the star's distance from the tiny parallax angle and the known Earth-orbit baseline.