Find tangent line to the curve, given a point:

f(x) = cos(x) – sin(x), (pi, -1).

f'(x) = – sin(x) – cos(x), which is what you got, Nicholas.

The slope of the tangent line at the given point is:

m = f'(pi) = – sin(pi) – cos(pi)

Trig refresher:

Given any point, P(x,y), define A(0,0), B(x,0), θ is angle between AB and AP, r = √(x^2+y^2).

We define (r cos(θ),r sin(θ)) = (x,y).

If (x,y) is on a Unit Circle centered at the origin, r = 1, so (cos(θ),sin(θ)) = (x,y).

Once around the Unit Circle is 2 pi radians, so pi radians is halfway around the Unit Circle starting from (1,0), moving CCW, and ending at (–1,0) = (cos(pi),sin(pi)).

Now we can find our slope.

m = f'(pi) = – sin(pi) – cos(pi) = – (0) – (–1) = 1

Using (pi, -1) and the point-slope form we get the tangent line:

y – (–1) = 1 (x – pi)

y = x – pi – 1

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