find an equation for the line tangent to curve at the point defined by the given value of t

Question

x= 3 sin t, y= 3 cos t, t= 3pi/4

Answer 1

When t=3(pi)/4, x=3sin(3(pi)/4)=3sin((pi)/4)=3sqrt(2)/2, and y=3cos(3(pi)/4)=-3sqrt(2)/2.

dy/dx*dx/dt=dy/dt, so dy/dx*3cos(t)=-3sint; dy/dx=-tan(t). This is the tangent, and at t=3(pi)/4 dy/dx=-1. The equation of the tangent has the form y=mx+c, where m=dy/dx=-1. To find c we substitute (3sqrt(2)/2,-3sqrt(2)/2) into the equation: -3sqrt(2)/2=-3sqrt(2)/2+c, from which c=0, and y=-x is the equation of the tangent when t=3(pi)/4.