Finding Optimal Elevation Angles for a Field Goal: A Kinematics Trig Problem

[GunnaSix]

Homework Statement

A football kicker can give the ball an initial speed of 25 m/s. What are the least and greatest elevation angles at which he can kick the ball to score a field goal from a point 50 m in front of goalposts whose horizontal bar is 3.44 m above the ground?

The Attempt at a Solution

I've worked it down to 3.44(cos a)^2 = 50(sin a)(cos a) - 19.6 , but I can't figure out how to solve this equation. None of the trig identities seem to help. Is this equation solvable (without a graphing calculator) or am I just approaching the problem the wrong way?

 

[resaypi]

try
2sin(a)cos(a) = sin(2a)
and
cos(2a) = 2cos(a)^2 - 1

 

[GunnaSix]

Made the subs, now I have 1.72 cos(2a) - 25 sin(2a) = -21.32 . What now?

 

[ideasrule]

Assuming you did everything correctly, you can use the identity cos(2a)=1-2sin^2(a) to get a quadratic equation.

 

[GunnaSix]

Wouldn't I have to either get everything in terms of (sin a) or everything in terms of (sin 2a) to solve as a quadratic?

 

[resaypi]

You can subsitute cos(2a) = (1-sin^2(2a))^0.5 and square the equation to get a quadratic equation. Solve for sin(2a).