Proving the Laws of Sines and Cosines

Lesson 1 of 2: The Proofs

In this lesson:

• Prove the Law of Sines using altitudes
• Prove the Law of Cosines using coordinate geometry
• See how both laws extend right-triangle trig to any triangle

Learning Objectives

By the end of this lesson, you should be able to:

1. State the Law of Sines and explain its geometric meaning
2. Prove the Law of Sines using the altitude method
3. State the Law of Cosines and explain how it generalizes the Pythagorean Theorem
4. Prove the Law of Cosines using coordinate geometry and the Pythagorean identity

SOH-CAH-TOA Has a Limit

A triangle has angles 50°, 60°, and 70°. One side is 10 cm.

• No right angle — so no hypotenuse
• SOH-CAH-TOA requires a right angle as its anchor
• We cannot use it here

How do we find the remaining sides and angles?

Law of Sines

What the Law of Sines Means

• Each fraction pairs a side with its opposite angle
• The ratio is the same for all three pairs in any triangle
• Equivalently:

Proving the Law of Sines — Step 1

Start with triangle . We want to show .

Proving the Law of Sines — Step 2

Draw altitude from perpendicular to . Two right triangles share the leg :

Proving the Law of Sines — Step 3

From the two right triangles, both expressions equal :

Divide both sides by :

Quick Check

In the Law of Sines, side pairs with which angle?

• A) Angle
• B) Angle
• C) Angle

Look at the triangle diagram — which angle does side face?

Proving the Law of Sines — Step 4

Repeat with an altitude from vertex :

In the two new right triangles:

Setting equal and rearranging:

Combining both results:

An Elegant Alternative Proof

From HSG.SRT.D.9, the area of triangle can be written three ways:

Divide every term by :

Example: Using the Law of Sines

Given: , , . Find .

Step 1: Set up the proportion

Step 2: Solve for

Spot the Error

A student writes:

What is wrong?

Hint: look at which side is paired with which angle...

The Law Includes Right Triangles

Let . Then , and the Law of Sines gives:

So — this is exactly SOH-CAH-TOA.

The Law of Sines does not replace right-triangle trig — it extends it.

Now: The Law of Cosines

The Law of Sines works when you know a side and its opposite angle.

But what if you know:

• Two sides + the angle between them (SAS)?
• All three sides (SSS)?

In both cases, you have no side paired with its opposite angle yet.

→ We need the Law of Cosines

Law of Cosines

All three equivalent forms:

Each form finds one side from the other two sides and their included angle.

The Correction Term

• If : → correction vanishes → Pythagorean Theorem
• If : → is shorter than
• If : → is longer than

Proving the Law of Cosines — Setup

Place at the origin, at , and at .

Distance Formula Gives

Side = distance from to :

Expand

Regroup by collecting the terms:

Apply the Pythagorean Identity

Substituting:

Quick Check

What does simplify to when ?

Write the simplified equation before advancing...

Pythagorean Theorem is a Special Case

When :

The Pythagorean Theorem is the Law of Cosines at .

The Law of Cosines generalizes every right-triangle result you've used.

The Minus Sign — Why It Matters

Watch out: The sign is always minus — never plus.

• The minus comes from in the distance formula
• When is obtuse: , so — gets longer
• Quick check: substitute — with a plus sign, the result wouldn't equal

Example 1 — SAS: Find the Missing Side

Given: , , . Find .

Spot the Error

A student writes:

What is the error, and why does it matter?

Hint: substitute into both versions...

Example 2 — SSS: Find an Angle

Given: , , . Find angle .

Rearrange the Law of Cosines to solve for :

Don't Stop at

After the Law of Cosines gives you , apply the inverse cosine:

The number is not an angle. It is the cosine of the angle.

• Always write both steps explicitly
• Report in degrees, not

Key Takeaways

✓ Law of Sines: — each side with its opposite angle

✓ Law of Cosines: — generalizes the Pythagorean Theorem

✓ Both proofs use only geometry you already knew

Side pairs with angle — never with angle or

Law of Cosines uses minus — traced to in the distance formula

After finding , always compute — the cosine is not the angle

Coming Up: Solving Triangles

Now that you can prove both laws, let's put them to work.

Lesson 2 covers:

• Choosing the right law for each case (AAS, ASA, SAS, SSS)
• Solving complete triangles step by step
• The ambiguous SSA case — and why it can have 0, 1, or 2 solutions

→ HSG.SRT.D.10, Lesson 2