Sound, the seasons, the power in your wall — all the same wave underneath.
The trig table ends at 90°. So what on earth is sin 150°?
Trig ratios set spinning on a circle — the language of everything that turns.
Experiment
Hands-on experiment
Predict first — the trig table stops at 90°. So what is sin 150°?
Spin the Ferris wheel
A point on a circle of radius 1 — its height (y-coordinate) IS the sine. Push past 90°.
0°
angle
0.000
height (sin θ)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Trig ratios were the surveyor's ruler, but they stop at 90°. The world keeps turning past 90° — wheels, planets, pendulums, waves. Rotation needed a language.
The solution: detach the ratios from the right triangle and put them on a circle. Redefine sin as the height of a point on the unit circle and cos as its horizontal position — every angle gets a value, and the values repeat once per revolution.
That 'repeating function' became the mathematics of waves. Sound, light, radio, AC power, heartbeats — every periodic phenomenon can be written as combinations of sines and cosines (Fourier's discovery).
Insight
Insights from the video
“The sine curve is the shadow of rotation.”
Watch a Ferris wheel from the side and each cabin only moves up and down. Record that vertical motion over time and you get exactly the sine curve — a wave is circular motion's one-dimensional shadow.
“Periodicity is nature's basic grammar.”
Earth's spin (a day), its orbit (a year), heartbeats, AC power (60Hz) — nature is full of repetition. Trig functions are the mathematics of repetition, which is why sine appears everywhere in physics.
Misconception
Common misconceptions
Sine needs a right triangle to be defined.
That was the 0°–90° era. In the extended definition, sin θ is 'the height (y-coordinate) of a point rotated θ around the unit circle' — no triangle required, and 150°, 1000°, or negative angles all get values.
The sine curve's wave shape is just how the graph happens to look.
The wave is rotation's shadow. Unroll the height of a circling point along a time axis and that curve appears — sine shows up in wheels, sound, and seasons because they are all rotation or repetition.
Formula
Writing it as math
What the Ferris-wheel experiment showed, in mathematical language.
Extended definition (unit circle)
Rotate θ around the circle of radius 1: the point's coordinates are (cos θ, sin θ). No triangle needed — every angle now has a value.
Periodicity
One full turn returns you home — which is why the graph repeats like a wave. This repetition is the heart of wave mathematics.
The general wave
A (amplitude) is the wave's height; ω (angular frequency) is the spinning speed. A sound's loudness and pitch are exactly these two knobs.
In Real Life
Where you meet it in real life
Sound and music
The note A4 is a sine wave vibrating 440 times per second. An instrument's sound is a sum of sine waves — timbre is the mix of overtones.
AC power
Your outlet's voltage sloshes as a 50/60Hz sine wave. A generator's rotation becomes a sine curve and travels to your home unchanged.
Seasons and day length
Across a year, day length traces a sine curve — the shadow of Earth's orbital rotation. Summer and winter are the curve's crest and trough.
Radio and wireless
Wi-Fi, 5G, radio — all wireless communication rides information on sine waves. Modulation is the art of manipulating A and ω.
Try Yourself
Test yourself
Q1What is sin 150°? Think on the unit circle.Show answer ▾
150° sits past 90°, mirroring 30° on the other side — its height equals sin 30° = 1/2. On the unit circle the 150° and 30° points share the same y-coordinate (sin 150° = 0.5).
Q2When is sin θ negative?Show answer ▾
When the point is on the circle's lower half — 180° < θ < 360°. The 'height' dips below the horizon, so it's negative — the extended definition holds negatives naturally.
Q3If the Ferris wheel spins twice as fast, how does the height graph change?Show answer ▾
The wave gets twice as tightly packed — y = sin(2ωt), half the period. The frequency doubled; as sound, that's one octave higher.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Trigonometric Ratios
Meet the ratios inside a right triangle first, and the extension onto the circle lands dramatically.
← Trigonometric Ratios labLeads to next
Vectors
The unit-circle point (cos θ, sin θ) is really an arrow of length 1 — leading to numbers that carry direction: vectors.
Go to the Vectors lab →Related
Labs worth exploring together
Related lab
Trigonometric Ratios
The 0°–90° ratio-table era — the ancestor of trig functions.
Go to the Trigonometric Ratios lab →Related lab