sin θ · Trigonometric Ratios
The angle fixes the side ratios — that's trigonometry.
The distance to a tree across the river — can you measure it without crossing?
Read trigonometric ratios through similarity.
The ancient astronomers who reckoned the distance to the Moon without leaving Earth.
Open the immersive version →A reading path containing this lesson: Measure without reaching · 3/3
Why does this question matter?
Ancient surveyors and astronomers faced 'distances you cannot walk': across the river, to the mountaintop, to the Moon — lengths that had to be measured without being reached.
The key was similarity. Right triangles with equal angles have identical side ratios at any size. So tabulate the ratios per angle, and one angle plus one distance computes every other side.
That table of ratios is trigonometry. From Hipparchus's chord tables to GPS satellites — the surveying revolution of 'calculate instead of measure' started here.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — keep the angle at 30° and make the right triangle twice as big. What happens to (opposite ÷ hypotenuse)?
Pick an angle, then change the size
Hypotenuses 5, 10, 20 — compare (opposite ÷ hypotenuse) across triangles 4× apart in size.
| hypotenuse | opposite | opp ÷ hyp |
|---|---|---|
| 5 | 2.50 | 0.500 |
| 10 | 5.00 | 0.500 |
| 20 | 10.00 | 0.500 |
All three triangles give exactly 0.500. Equal angles mean equal shape (similarity), and equal shape means equal ratios. Size cannot touch the ratio — it is the fingerprint of the angle alone.
Read the idea from another angle
A trig ratio is the fingerprint of an angle, not a length.
Fix an angle and the right triangle's shape is fixed; fix the shape and the ratios are fixed. sin 30° = 1/2 everywhere in the universe — the angle's own fingerprint.
Thales measured the pyramid with a shadow.
A stick's shadow and the pyramid's shadow — same moment, same sun angle, so the triangles are similar. That insight from 2,600 years ago is trigonometry's prototype.
Now read the definitions with their conditions
Name the unchanging ratios you confirmed in the experiment, and you get these.
Definitions
Three ratios from a right triangle's three sides. Same angle, same values — size is irrelevant.
The height-measuring formula
Know the baseline (distance) and the elevation angle, and the height falls out — the one-liner behind river, building, and mountain surveying.
The family relations
tan is the ratio of sin to cos, and sin²+cos²=1 is the Pythagorean theorem in disguise — the three are one family.
Make one more distinction
A bigger triangle has a bigger sine.
Sine is a ratio, not a length. In similar triangles all sides scale together, so the ratio stays put — sin 30° is exactly 1/2 in a palm-sized triangle and in a mountain-sized one.
sin, cos, tan are three unrelated formulas.
They're three ratios read off one right triangle. tan = sin/cos, and sin²+cos²=1 (Pythagoras in disguise) — know one and the others follow.
Where else can this idea help?
Surveying and map-making
Triangulation — one baseline plus angles maps an entire country. It was the skeleton of cartography from the Age of Exploration until GPS.
Aircraft landing angles
Airliners approach runways on a 3° glideslope. tan 3° ≈ 0.052 — descending 52m for every kilometer forward.
Ladder safety codes
Industrial safety standards set ladders at 75° — wall distance : height = 1 : 4 (tan 75° ≈ 3.7). Trigonometry written into law.
Field of view in games and graphics
A 3D camera's FOV, the angle a character turns to face a target — rotation and aiming in game engines are trig computations throughout.
Explain it in your own words
Try another question
Q1From 50m away, the angle up to a building's roof is 45°. How tall is the building?Show answer ▾
Height = 50 × tan 45° = 50 × 1 = 50m. 45° is the special angle where base and height coincide (tan 45° = 1).
Q2sin 30° = 1/2. In a right triangle with hypotenuse 8, how long is the side opposite the 30°?Show answer ▾
opposite/hypotenuse = 1/2, so opposite = 8 × 1/2 = 4. Know the ratio and one multiplication gives the side — the power of the ratio table.
Q3You ski down a 100m slope inclined at 30°. How far did you travel horizontally?Show answer ▾
Horizontal = 100 × cos 30° = 100 × (√3/2) ≈ 86.6m; vertical drop = 100 × sin 30° = 50m. One angle reveals both displacements.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Measure without reaching · 3/3
Previous concept
Pythagorean Theorem
Know the length relations of a right triangle first, and the ratio relations come into view.
← Pythagorean Theorem labLeads to next
Trigonometric Functions
Release the ratios onto a circle and they pass 90°, reach 360°, and become waves — the surveyor's ruler becomes the language of rotation.
Go to the Trigonometric Functions lab →Related lab
Pythagorean Theorem
The true identity of sin²+cos²=1 — trigonometry's foundation.
Go to the Pythagorean Theorem lab →Related lab