Can a basic physics problem give you insight into math?
For example, mathematically
which is easy to verify using the angle addition formula.
I came across this formula while solving a simple problem in statics.
Imagine the classic “block on an inclined plane”. Gravity () pulls the block down, and you push () on it sideways, like this:
What is the minimum coefficient of static friction to keep the block stationary? In order to calculate this, we need to know the component of force parallel to the plane.
First look at gravity. We want to find the green component .
Let’s say the positive direction is to the right. Then gravity is pulling backwards some, so is negative. I know it’s either a sine or cosine of , and in the limit as I see that , so
Then we look at the pushing force.
A similar procedure gives
So the total force in the direction of the ramp is
In the special case where the force is
Now we will find this component of the force another way. We start by tip-to-tail adding the force of gravity and the force of the push. They’re at right angles, and assuming they’re equal in magnitude we get a resultant force with length that bisects the angle between the gravity and pushing forces.
The angle between this resultant force and the plane is .
The component of this force along the plane is then the cosine of , so the force along the direction of the plane is
and since it’s the same quantity we calculated before, we have
There is no physics in this calculation, but if you had simply asked me to write as a single trig function, I wouldn’t have thought to approach it like this.