y-Component from Magnitude and Angle

The vertical partner of the x-component: |v| sin θ is how much of the arrow points up. A projectile launched at 40 m/s and 30° leaves the muzzle with 40 × sin 30° = 20 m/s of upward speed, and that number alone — not the 40 — decides how high and how long it flies. Together the pair (|v| cos θ, |v| sin θ) is the machinery that lets you replace one awkward diagonal with two independent one-dimensional problems, which is the reason vectors earn their place in physics at all.

Solving for the angle uses arcsin, whose principal branch runs only from −90° to 90°. A vector at 150° has the same y-component as one at 30°, so formula.expert returns the first-quadrant answer; if you know the vector leans left, its true direction is 180° minus what you get. When the whole vector is horizontal, sin θ = 0 and no y-component can tell you how long it is.