Question: Why do θ = 30° and θ = 60° give the same range for the same launch speed u?

Chain:

$R = \frac{u^2\sin 2\theta}{g}$

→ For θ = 30°: $R_{30} = \frac{u^2\sin(60°)}{g} = \frac{u^2 \times 0.866}{g}$

→ For θ = 60°: $R_{60} = \frac{u^2\sin(120°)}{g} = \frac{u^2 \times 0.866}{g}$

→ $\sin(2 \times 30°) = \sin 60°$ and $\sin(2 \times 60°) = \sin 120°$

→ But $\sin 60° = \sin(180° - 60°) = \sin 120°$ (supplementary angles have equal sines)

→ Therefore $R_{30} = R_{60}$ ✓

Why heights differ:

$H = \frac{u^2\sin^2\theta}{2g}$

→ $H_{30} = \frac{u^2\sin^2 30°}{2g} = \frac{u^2(0.25)}{2g}$

→ $H_{60} = \frac{u^2\sin^2 60°}{2g} = \frac{u^2(0.75)}{2g}$

→ $\frac{H_{60}}{H_{30}} = \frac{0.75}{0.25} = 3$

→ The steeper angle (60°) converts more initial speed into vertical motion → rises 3× higher

Takeaway: Equal ranges arise from equal $\sin(2\theta)$ values (supplementary 2θ angles). But $\sin^2\theta$ values differ — hence unequal heights. This is a pure trigonometric result, not a coincidence.