Why ranges are equal at 30 degree and 60 degree?

Answer 1

this is due to effect of sin2(theta). as sin2(theta) will repeat its value for =90-theta

as here

theta=30 so

for 90-30=60

sin2(30)=sin2(60)

so for pair of projection angles of two projectiles as(theta,90-theta) , they will have same ranges

i.e

theta=10

and 90-10=80

sin2(10)=sin2(80)

Answer 2

At 30 degrees and 60 degrees, the sine values are equal. Since the range of a projectile depends on the initial vertical velocity (which is influenced by the sine of the launch angle), the ranges are equal at these angles as the vertical component of the initial velocity remains the same.