In this lesson:
By the end of this lesson, you should be able to:
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.
Two triangles are similar if they have the same angles (same shape, possibly different size).
For right triangles, the AA criterion says:
Conclusion: All right triangles with the same angle are similar to each other.
All three triangles have the same acute angle — so they are all similar.
The ratio opposite/hypotenuse is the same in all three triangles.
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.
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...
Since every right triangle with angle produces the same ratios:
These ratios are functions of the angle. We give them names.
This is the foundation of trigonometry.
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
For a right triangle with acute angle :
Important: Opposite and adjacent are relative to a specific angle — they change if you switch to the other acute angle.
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.
Identify the angle first — then the sides.
Sine of angle (abbreviated sin):
Cosine of angle (abbreviated cos):
Memory aid — SOH-CAH:
Tangent of angle (abbreviated tan):
Memory aid — TOA:
Important: Tangent compares the two legs — it can be any positive number, not just between 0 and 1.
Full mnemonic: SOH-CAH-TOA
Which trigonometric ratio uses the adjacent side and the hypotenuse?
Can you name it before checking?
These three numbers are properties of angle — same in every right triangle with this 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.
Three triangles, three different sizes — all with 40° — and the ratios are identical.
Calculators return the ratio for any angle:
Make sure your calculator is in degree mode for these problems.
Given this right triangle with angle A at the bottom-left:
Task:
Work through each step before advancing.
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.)
For each right triangle, label opposite, adjacent, and hypotenuse for the given angle, then compute all three ratios.
Work independently, then check your answers on the next slide.
Problem 1 (opp=8, adj=15, hyp=17):
Problem 2 (opp=5, adj=12, hyp=13):
Problem 3 (opp=7, adj=24, hyp=25):
✓ 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
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:
The same three ratios, now applied to solve problems.
Click to begin the narrated lesson
Understand trigonometric ratios