Listen, beta, it's about which angle is perpendicular. The sine comes from the angle between the line and the plane's *normal*, which is 90 degrees shifted. The cosine of that angle with the normal gives the sine for the plane itself.
I see it like my embroidery hoop—the needle (line) goes deep at an angle, but its shadow (projection) on the cloth (plane) is measured differently. My old geometry book, stained with turmeric, has this very diagram in the margin.
We use sine because the real relationship is with the upright normal, like how a bride stands straight while her lehenga flows along the floor.
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