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

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Answer 1 · 2017-02-03T12:18:01

# y = pix-3pi #

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]}