Complementary angles range

Intermediate

Launch angles θ and 90°−θ give the same range.

Key relationprojectile

Because sin2θ = sin(180° − 2θ), launch angles θ and 90° − θ give the same range, with the higher angle having more height and time.

Worked example

30° and 60° both reach 0.866u²/g.

Common mistake

They share range, not flight time or height.

In more detail

Two launch angles that add up to 90 degrees, such as 30 and 60, give the same horizontal range, because sin 2θ is the same for both.

Two golfers hitting with different loft angles can achieve the same distance.
Ballistics analysts use complementary angles when calculating shot trajectories.
A cricketer hitting a ball at a low angle or a high angle can land it at the same distance, which is useful when choosing a shot.

Why do complementary angles give the same range?

Because sin 2θ equals sin(180 degrees minus 2θ), which is the same for θ and 90 degrees minus θ.

Which one reaches higher?

The steeper angle gives the higher trajectory and longer time of flight.

Are the two launch angles equally good?

They give the same range, but the high angle gives a longer time in the air and a higher trajectory.