Watch the circle draw the wave in real time — sin and cos are just circular motion seen sideways
MAT 172 — Visual Proof
Mathematicians didn't invent sin and cos as wavy graph functions. They invented them as ratios inside a circle. The wave shape is what happens when you take circular motion and ask "what does just the height look like over time?" One lap of the circle, one full wave. That's why the period is 2π.
Live animation — the circle drawing the wave
Angle θ
0°
sin(θ) = y
0.000
cos(θ) = x
1.000
tan(θ) = y/x
0.000
Speed:1×
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What you're seeingThe dot on the circle is the source. The wave on the right is a recording of that dot's height (for sin) or horizontal position (for cos) as time passes. They are the same information — one circular, one unrolled into a line.
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Lab crossoverThink of a centrifuge vial. As it spins, pick one point on the rim and watch only its vertical position — that traces a sine wave. The spin is the circle. The height-over-time readout is the wave. Same motion, two ways of looking at it.
Four things to watch for
Zero crossings
The wave crosses zero exactly when the dot crosses the x-axis — at 0°, 180°, 360°. Watch the dot hit the left and right edges of the circle.
Peaks and troughs
The wave peaks at 1 when the dot reaches the very top of the circle (90°). It troughs at −1 at the bottom (270°).
Sin vs cos offset
Switch to "both" — cos starts at 1 while sin starts at 0. They're the same wave, just a quarter turn (π/2) apart.
Period = one full lap
One complete wave cycle takes exactly as long as one full trip around the circle. That distance is 2π — that's why the period is 2π.
The key insight
Every property of the sine and cosine graphs — amplitude, period, zeros, shape — is just a fact about circles wearing a different outfit. The amplitude is always 1 because the circle has radius 1. The period is 2π because that's one full revolution. The zero crossings at nπ because that's where the dot crosses the horizontal axis.
The ladder you imagined falling against the wall? The height of the top of the ladder at every angle IS the sine wave. Watch the animation: that dot is the top of the ladder.
What each color means
Purple dot — the point rotating on the unit circle
Teal line — the sine wave (y-height over time)
Coral line — the cosine wave (x-width over time)
Dashed line — connects the circle dot to the wave dot
Key connections
sin = height
The y-coordinate of the dot on the circle. How high up the dot is at any angle.
cos = width
The x-coordinate of the dot. How far right or left the dot is at any angle.
Period = 2π
One full lap of the circle = 2π radians = one complete wave cycle. Not a coincidence — it IS the same thing.
Amplitude = 1
The circle has radius 1. The dot never goes higher than 1 or lower than −1. So sin and cos are always in [−1, 1].
Phase shift π/2
Sin and cos are the same wave — cos just starts a quarter-turn ahead. cos(θ) = sin(θ + π/2).
Watch for these moments
At θ = 0°: dot is at right. sin = 0, cos = 1. Wave starts at zero (sin) or peak (cos).
At θ = 90°: dot is at top. sin = 1, cos = 0. Sin peaks, cos crosses zero.
At θ = 180°: dot is at left. sin = 0, cos = −1. Both waves cross the axis.
At θ = 270°: dot is at bottom. sin = −1, cos = 0. Sin troughs, cos crosses zero.
Switch to "both" and pause at 45° — sin and cos are equal (both √2/2 ≈ 0.707). That's why tan(45°) = 1.