Position of Point on Circle → Values Plotted on Wave Chart
Guide Features: Click "Guides Off" to hide advanced educational features:
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Angle Sweep: Green sector shows θ growing from 0, changing color each full rotation
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Arc Length: Yellow dashed arc shows θ = arc length on unit circle
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Triangle Highlight: Fading triangles show how each wave point is born from geometry
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Projection Lines: Dashed lines connect circle positions to wave positions
The Shadow Concept: Imagine a light shining from the left onto the rotating point.
The sine wave is literally the shadow cast by this point as it moves around the circle!
Toggle "Show Shadow" to see this effect clearly.
Phase Shift: A phase shift moves a wave left or right along the horizontal axis.
When you click "Show Phase Shift", the blue cosine wave shifts by π/2 radians (90°) to align with the red sine wave.
This proves that cos(θ + π/2) = sin(θ) - they're the same wave, just offset in time!
90° Angle: The "Show 90° Angle" button reveals the special relationship between sine and cosine.
It shows a second point that's always 90° ahead of the main point. Notice how when the main point is at the top (max sine),
the 90° point is at the left (zero cosine). This visual proves why cosine leads sine by exactly 90°!
Tangent Line: The "Show Tangent" button displays the geometric definition of tangent on the unit circle.
A vertical line at x=1 intersects with the extended radius line. The y-coordinate of this intersection point is tan(θ).
Notice how tan(θ) = sin(θ)/cos(θ) - it's literally the slope of the radius line! When cos(θ) = 0 (at 90° and 270°),
tangent becomes undefined (infinite), which you can see as the orange line extending to infinity.
💡 Key Insight: "Each point on the wave is born from a triangle. The angle θ becomes distance along the wave, and the triangle's legs become the wave's height."
🔺 Ready for SOHCAHTOA?
Master the triangle relationships with our interactive
SOHCAHTOA Explorer
- perfect for understanding sin, cos, and tan in right triangles!