Law of Sines

Trigonometry — Non-Right Triangle Solutions

In this lesson:

• Apply the Law of Sines to find missing sides and angles
• Handle the ambiguous case with confidence
• Solve ACT-style problems efficiently

What You Will Learn Today

After this lesson, you will be able to:

1. State the Law of Sines formula
2. Solve for missing sides using AAS or ASA
3. Solve for missing angles using SSA
4. Analyze the ambiguous case for possible solutions

What if the Triangle Has No Right Angle

You know SOH-CAH-TOA works for right triangles:

But most real-world triangles are not right triangles.

We need a tool for oblique triangles — and that's the Law of Sines.

The Law of Sines Relates All Sides

When to Apply the Law of Sines

You need one complete side-angle pair plus one more piece:

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

Not enough? SAS or SSS requires the Law of Cosines instead.

AAS Example: Find the Missing Side First

In : , ,

Step 1: Find angle

Step 2: Set up the proportion for side

AAS Example: Solve for Both Sides

From previous slide: ,

Solve for :

Solve for :

Quick Check on Identifying Complete Pairs

In : , ,

Which is the complete side-angle pair?

Think about it before advancing...

ASA Example: Same Strategy Applies Here

In : , ,

Step 1: Find angle

Step 2: Solve for side

Why SSA Creates an Ambiguous Situation

Fix angle and side . Side swings like a compass arc.

Three Possible SSA Outcomes to Consider

Given angle , side (opposite), and side (adjacent):

1. No triangle: — impossible
2. One triangle: is valid, but exceeds
3. Two triangles: Both and are valid

SSA Two-Solution Example: Find Angle B

In : , ,

Set up:

SSA Two Solutions: Check the Supplement

From previous:

Check supplement:

Valid? — Yes!

Two triangles exist:

• Triangle 1: ,
• Triangle 2: ,

Quick Check: Does a Second Triangle Exist

In : , , so

Does produce a valid triangle?

Check: does ?

SSA Checklist: Follow These Four Steps

For every SSA problem, follow this process:

1. Solve for using the proportion
2. If → no triangle
3. If → find
4. Check: does also give a valid triangle?

SSA No-Solution Example: Sin B Exceeds One

In : , ,

Since : No triangle exists.

Side is too short to reach the opposite side.

ACT Practice: Straightforward AAS Problem

In : , ,

Find side .

A) 11.3   B) 14.0   C) 16.8   D) 17.3

Solve before advancing...

ACT Practice: AAS Problem Solution Revealed

Step 1:

Step 2: Set up the proportion

Step 3: Solve

Answer: C

ACT Practice: How Many Triangles Possible

In : , ,

How many distinct triangles can be formed?

A) 0   B) 1   C) 2   D) Cannot be determined

Use the SSA checklist...

ACT Practice: Triangle Count Solution Revealed

— valid ()

— valid ()

Answer: C) Two triangles

Key Takeaways and Common Mistakes

• Formula:
• Pair each side with its opposite angle
• Step 1: find the missing angle first

Watch out:

• SSA: check if also works
• SAS/SSS → use Law of Cosines instead
• Calculator in degree mode

Coming Up Next in Trigonometry

Up next: Law of Cosines

• Handles SAS and SSS cases the Law of Sines cannot
• Generalizes the Pythagorean theorem to all triangles
• Completes your toolkit for solving any triangle