Question

What is the slope of the polar curve `f(theta) = sectheta - csctheta` at `theta = (3pi)/8`?

Answer 1

`m = 6.757`

Explanation:

The slope of the tangent line of a function at a point is equal to the derivative of the function at that point.

With that being said, let's take the derivative

`(df)/(d theta) [f(theta) = sectheta - csctheta]`

The derivative of `sectheta` is `tanthetasectheta`:

`f'(theta) = tanthetasectheta - d/(d theta) [csctheta]`

The derivative of `csctheta` is `-cotthetacsctheta`:

`ul(f'(theta) = tanthetasectheta + cotthetacsctheta`

Or, in terms of `sin` and `cos`:

`ul(f'(theta) = (sintheta)/(cos^2theta) + (costheta)/(sin^2theta)`

Now, to find the slope at the point `(3pi)/8`, we plug it in for `theta`:

`m = (sin((3pi)/8))/(cos^2((3pi)/8)) + (cos((3pi)/8))/(sin^2((3pi)/8)) = color(blue)(2sqrt(10+sqrt2)`

`= color(blue)(ulbar(|stackrel(" ")(" "6.757" ")|)`