Wow, a real life formula that uses the double angle trig. identities! As you can see, the distance a projectile will travel is a function of: velocity, gravity, and the launch angle.

First, a quick fraction review: First, recall that is a lot smaller than . Conversely, we can also agree that is a lot smaller than . ie: The bigger the denominator (and/or smaller the numerator), the lower the value of the (positive) fraction.

So, since v is in the numerator, the distance traveled *(d)* increases directly with velocity (in a big way, since it’s squared) Next, since g is in the denominator, the distance traveled decreases as gravity increases. (Makes sense, right?)

This is a graph of all Sin(x) values from* *0 to 360. The x-axis is divided into quadrants (0, 90, 180, 270, 360). Notice in the graph that Sin(x) rises from 0 to 1 as *x* rises from 0 to 90 degrees. Then, it drops from 1 back to 0 as* x* rises from 90 to 180 degrees.

Refer back to the double angle in the original formula up top. So, as *x *rises from 0 to 45 degrees, *2x* actually rises from 0 to 90, and the value is increasing. But, as *x* continues to rise from 45 to 90 degrees, *2x *rises from 90 to 180, which means the value is now decreasing.

So, what’s the ideal angle to throw something? The one that maximizes the value of , since it’s a multiplier in the projectile formula. Well, as you can see in the graph, Sin(90) = 1, the highest possible value for Sin(x). So, the ideal launch degree is x = 45 (which puts 2x at 90).

So, now you know *why*** **the ideal angle in these video games is 45 degrees, and have an inkling of

*programmers create classic games like these:*

**how**