slope of the tangent line to the curve given by x=sin(y) at the point (0,3π) ?

Answer 1

If you are going to graph x = sin(y), it will not give you a function because a single vertical line test will intersect multiple points on the graph. However, y= sin^-1(x) is. So let's get the derivative of this:

dy/dx = 1/√(1-x²).

Since the original graph has multiple points at x=0, then we have to consider the derivative to be:

dy/dx = 1/±√(1-x²)

at (0,3π), you will see that the slope of the tangent line is going down. Therefore:

dy/dx = 1/±√(1-0²)

dy/dx = ±1 (the slope is going down so we eliminate the positive)

dy/dx =-1

Therefore the slope of the tangent line is -1.