How can I show that the range of a projectile will be the same for two complementary angles using trig identities?

Question

How can I show that the range of a projectile will be the same for two complementary angles using trig identities?

Answer 1

R = v₀²sin(2θ)/g = R(θ)

 

Let φ = 90° - θ be the complementary angle

 

R(φ) = v₀²sin(2φ)/g

 

sin(2φ) = 2sinφ cosφ = 2sin(90° - θ) cos(90° - θ) = 2cosθ sinθ = sin(2θ)

 

sin(2φ) = sin(2θ)

 

Therefore, 

 

R(φ) = R(θ) if φ = 90° - θ