Building up to a ray of light
The radiometry ladder
Five quantities, each one the previous divided by something. Energy becomes flux when you divide by time, flux becomes irradiance when you divide by area, and irradiance becomes radiance when you divide by solid angle. That last rung is what a ray of light actually is. Climb the ladder on the right, and tilt the beam to see where the cosine in the rendering equation comes from.
A beam meeting a surface
Drag up and down to tilt the beam
The ladder
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Beam cross-section
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Footprint on the surface
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cos θ
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Irradiance at the surface
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Controls
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Where the cosine in the rendering equation comes from. Tilt the beam and the light in it does not change — the same flux, the same photons. What changes is the area they land on: at 60° the footprint is twice the beam's own cross-section, so every square metre receives half as much. That is the entire content of
E = Φ·cosθ / A, and it is the same cosθi sitting inside the integral of the rendering equation. It is not an optical effect or an approximation; it is geometry. A surface facing away from a light is dim for exactly the reason a shadow on the ground stretches out at sunset.