# Arc Length for Polar Functions

Polar functions are a special case of parametric functions. These are just equations disguised as nasty trigonometric functions. It's sort of like they're wearing intricate Halloween costumes, like a stick figure man costume made of black cloth and glow sticks. We can't tell who it is, but we can do our best to use our knowledge about a person to figure it out.

The polar function *r* = *f*(*t*) can be parametrized as

*x*(*t*) = *r* cos *t* = *f*(*t*) cos *t*

*y*(*t*) = *r* sin *t* = *f*(*t*) sin *t*

We're using *t* instead of *θ* so we can talk about instead of , to be consistent with the earlier discussion of parametric functions. To find and , we have to apply the product rule carefully.

### Sample Problem

Find the arc length for one petal of the polar function *r* = sin(3*t*).

Answer.

The function looks like

The first petal corresponds to the interval . We get

*x*(*t*) = *r* cos *t* = sin(3*t*)cos *t*

*y*(*t*) = *r* sin *t* = sin(3*t*)sin *t*.

Using the product rule gives us

We use the arc length formula and get

It looks complicated and drawn out, but we can always just use a calculator to solve the integral. If our teacher asks us to solve this one, they're looking to torture us.

Sometimes, a slightly modified costume simplifies the calculations. We can use the identity

sin^{2} *t* + cos^{2} * t* = 1

to get a nicer formula for the arc length of a polar function.

If we want to make the formula look more like it goes with polar functions, we can put *θ* in place of *t*:

This formula is nice to keep handy, but we probably don't need to memorize it unless we need to do a lot of polar arc length problems.