Trigonometric Ratios: Foundations — Lesson 1

Understanding Why Trig Works

In this lesson:

• Explain why similar right triangles produce consistent side ratios
• Define sine, cosine, and tangent for any acute angle
• Understand trig ratios as properties of angles, not triangles

What You Will Learn Today

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

1. Explain how similarity of right triangles leads to consistent side ratios for a given angle
2. Define sine, cosine, and tangent ratios for acute angles in right triangles
3. Understand that trigonometric ratios depend only on the angle, not the triangle's size
4. Recognize the connection between trigonometric ratios and the concept of similarity

How Can Angles Reveal Distances?

An ancient problem: How tall is that mountain? How wide is that canyon?

Surveyors in ancient Greece measured distant objects by measuring angles — without ever going there.

The key insight: If you know an angle and one distance, angles reveal proportions.

Today we'll discover exactly why angles carry this power.

Similar Triangles: A Quick Review

Two triangles are similar if they have the same angles (same shape, possibly different size).

For right triangles, the AA criterion says:

• All have a right angle (90°)
• Share the same acute angle
• Third angle automatically equals

Conclusion: All right triangles with the same angle are similar to each other.

Three Different Sizes, One Angle

All three triangles have the same acute angle — so they are all similar.

The Magic: Ratios Don't Change

The ratio opposite/hypotenuse is the same in all three triangles.

Why? Because Similarity Guarantees It

Proof sketch:

Let and both have a right angle and acute angle .

By AA, .

Corresponding sides are proportional:

This holds for any two right triangles with angle — so the ratio is the same in all of them.

Quick Check: Similarity and Side Ratios

Two right triangles both have an acute angle of 40°.

Are they similar? What does that tell you about their side ratios?

Think before advancing...

Trig Ratios: Properties of Angles, Not Triangles

Since every right triangle with angle produces the same ratios:

• The ratio opposite/hypotenuse depends only on
• The ratio adjacent/hypotenuse depends only on
• The ratio opposite/adjacent depends only on

These ratios are functions of the angle. We give them names.

This is the foundation of trigonometry.

Transition: Let's Name the Ratios

We know the ratios are constant for a given angle.

Now let's label the sides and define the three ratios formally.

From consistent ratios → named functions: sine, cosine, tangent

Labeling Sides: It Depends on the Angle

For a right triangle with acute angle :

• Hypotenuse: the longest side, opposite the right angle (always the same)
• Opposite: the side across from angle
• Adjacent: the side next to angle (that isn't the hypotenuse)

Important: Opposite and adjacent are relative to a specific angle — they change if you switch to the other acute angle.

Quick Check: Opposite and Adjacent

A right triangle has angles A (bottom-left), B (bottom-right, right angle), and C (top).

The vertical leg is "opposite" for angle A.

If you switch to angle C instead — what is the vertical leg now?

(Opposite? Adjacent? Think about which side is across from C.)

The vertical leg is now adjacent to angle C — it sits next to C, not across from it.

Reference: Right Triangle Side Labels

Identify the angle first — then the sides.

Sine and Cosine: The First Two Definitions

Sine of angle (abbreviated sin):

Cosine of angle (abbreviated cos):

Memory aid — SOH-CAH:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse

Tangent: The Third Trig Ratio

Tangent of angle (abbreviated tan):

Memory aid — TOA:

• Tangent = Opposite / Adjacent

Important: Tangent compares the two legs — it can be any positive number, not just between 0 and 1.

Full mnemonic: SOH-CAH-TOA

Quick Check: Which Ratio Uses These Sides?

Which trigonometric ratio uses the adjacent side and the hypotenuse?

Can you name it before checking?

• A) Sine
• B) Cosine ← CAH: Cosine = Adjacent / Hypotenuse
• C) Tangent

Example: 3-4-5 Triangle, All Three Ratios

These three numbers are properties of angle — same in every right triangle with this angle.

Trig Ratios as Functions of the Angle

These aren't just formulas for one triangle — they're functions:

Input: any acute angle
Output: a number (the ratio)

The same angle always produces the same output — similarity guarantees it.

Same Angle, Same Ratios — Always

Three triangles, three different sizes — all with 40° — and the ratios are identical.

Using a Calculator for Trig Values

Calculators return the ratio for any angle:

• (exactly — opposite is half the hypotenuse)
• (adjacent and opposite are equal when )
• (opposite is longer than adjacent)

Make sure your calculator is in degree mode for these problems.

Your Turn: Label and Compute

Given this right triangle with angle A at the bottom-left:

• The vertical leg = 5
• The horizontal leg = 12
• The hypotenuse = 13

Task:

1. Identify: which side is opposite angle A? Adjacent? Hypotenuse?
2. Compute , , and

Work through each step before advancing.

Quick Check: Why Can tan Exceed 1?

Think about this:

Sine and cosine are always between 0 and 1 for acute angles.

Why can tangent() be greater than 1?

Hint: What sides does tangent compare?

(Tangent = opposite / adjacent — when opposite > adjacent, tan > 1.)

Practice: Compute All Three Ratios

For each right triangle, label opposite, adjacent, and hypotenuse for the given angle, then compute all three ratios.

1. Angle , sides: opp = 8, adj = 15, hyp = 17
2. Angle , sides: opp = 5, adj = 12, hyp = 13
3. Angle , sides: opp = 7, adj = 24, hyp = 25

Work independently, then check your answers on the next slide.

Practice: Check Your Answers Here

Problem 1 (opp=8, adj=15, hyp=17):

• , ,

Problem 2 (opp=5, adj=12, hyp=13):

• , ,

Problem 3 (opp=7, adj=24, hyp=25):

• , ,

Key Takeaways: What You Now Know

✓ All right triangles with the same acute angle are similar (AA criterion)

✓ Similarity guarantees constant side ratios — the foundation of trig

✓ opp/hyp, adj/hyp, opp/adj

✓ SOH-CAH-TOA — always identify your angle and label sides first

Watch out: Opposite and adjacent are relative to your angle — switch angles, and they swap

Watch out: Making the triangle bigger does NOT change sine, cosine, or tangent

Watch out: tan() can exceed 1 — it's opposite/adjacent, not a fraction of the hypotenuse

Coming Up Next: Solving Right Triangles

Lesson 2 of 2: Trigonometric Ratios — Solving Right Triangles

You now know what the ratios mean and why they work.

Next, you'll use them to:

• Find unknown sides given an angle and one side
• Find unknown angles given two sides (using inverse functions)
• Solve real-world problems: ladders, ramps, angles of elevation

The same three ratios, now applied to solve problems.