Lesson 7.2

The Law of Sines

The Law of Sines connects every angle to its opposite side through a single proportion — perfect for AAS and ASA cases.

Introduction

Given an angle–side pair plus one extra piece of information, the Law of Sines sets up a proportion that solves for any missing side or angle.

Past Knowledge

Oblique triangle classification (7.1), angle-sum property, cross-multiplication.

Today's Goal

Apply the Law of Sines to solve AAS and ASA triangles.

Future Success

The ambiguous case (7.3) extends this law to the trickiest SSA scenario.

Key Concepts

The Law of Sines

Equivalently:

When to Use

  • AAS: Two angles + non-included side
  • ASA: Two angles + included side

Always Find the Third Angle First

Use to get all three angles before setting up proportions.

Worked Examples

Basic

AAS — Finding a Side

Given: . Find .

Step 1:

Step 2:

Step 3:

Answer:

Intermediate

ASA — Full Solution

Given: . Find all parts.

Step 1:

Step 2:

Step 3:

Answer:

Advanced

Finding an Angle

Given: . Find .

(possibly also — see Lesson 7.3)

Answer:

Common Pitfalls

Calculator in Wrong Mode

If the problem uses degrees, your calculator must be in degree mode. A radian-mode calculation gives completely wrong results.

Real-Life Applications

Cell Tower Triangulation

When your phone connects to multiple cell towers, the network uses angle measurements and known tower distances to triangulate your position using the Law of Sines.

Practice Quiz

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