Finding the angle for maximum range with a constant initial velocity

[Failshire]

Homework Statement

According to the range equation (see below), what angle would give the maximum range for a given initial velocity? Mathematically explain your hypothesis.

Homework Equations

The equation given in our book for range:
R=(V$$_{o}$$cos$$\Theta$$)*(2V$$_{o}$$sin$$\Theta$$)/g

The equation given for range with same initial height and final height
R=Vo^2 sin(2$$\Theta$$)/g

The relevant identity of Sin:
sin(2$$\Theta$$)=2sin$$\Theta$$2cos$$\Theta$$

The Attempt at a Solution

This is not so much about math as it is about understanding the math, which is unfortunately my weakness. So, please excuse any confusion or ignorance, but this is my thought process so far along with a website that has pretty much explained my question but that I am unable to understand.

-The goal is to find the angle at which range is maximum, for the same velocity. Range is X-Xinitial. What I need to do is be able to make an equation that let's me figure out what angle will make range as big as it can be.

This is where I'm stuck. I found an awesome piece in Wired magazine that does this equation for me, but I'm completely at a loss to understand what its doing. The piece tells me that the biggest sin can be is 1, but why? How do I know that sin can never be bigger than 1? When I set sin to 1, and use the identity to solve for the angle theta, I know it will give me 45 degrees.

I suppose its not so much a homework question but a theory question. I have the answer to the homework, but I'm completely at a loss to understand why it works that way and would benefit from a deeper knowledge (because knowing is half the fun)

I hope this isn't too rambly. Theory homework is rough!


Link to wired article: http://www.wired.com/wiredscience/2010/09/maximum-range-in-projectile-motion/

 

[rock.freak667]

This is where I'm stuck. I found an awesome piece in Wired magazine that does this equation for me, but I'm completely at a loss to understand what its doing. The piece tells me that the biggest sin can be is 1, but why? How do I know that sin can never be bigger than 1? When I set sin to 1, and use the identity to solve for the angle theta, I know it will give me 45 degrees.

Because the nature of sin2θ or sin(nθ) [n=any real number] is such that its max. value is 1 and its minimum value is -1.

You can see this if y=sin2θ, dy/dθ= 2cos2θ, d²y/dθ²=-4sin2θ

For a stationary point, dy/dθ = 0 or 2cos2θ=0. Meaning that θ=45° (in this case θ varies from 0° to 90°)

when θ=45°, d²y/dθ² = -ve meaning that θ=45° makes 'y' maximum.