Question

What is the equation of the tangent line of `f(x)=(sinpix)/(cospix)` at `x=3`?

Answer 1

`y = pix-3pi`

Explanation:

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.

We have:

`f(x)=sin(pix)/cos(pix)`
`\ \ \ \ \ \ \ = tan(pix)`

Then differentiating wrt gives us:

`f'(x) = pisec^2(pix)`

When `x = 3 =>`

`f(3) \ \= tan(3pi) \ \ = 0`
`f'(3) = pisec^2(3pi)=pi`

So the tangent passes through `(3,0)` and has gradient `pi` so using the point/slope form `y-y_1=m(x-x_1)` the equation we seek is;

`y-0 = pi(x-3)`
`:. y = pix-3pi`

We can confirm this solution is correct graphically:
graph{(y-tan(pix))(y-pix+3pi)=0 [0, 4, -5, 5]}