Find the Cartesian equation of the line tangent to r = sin(theta) at theta= -pi/3.

Answer 1

I can get you started, but it will take me some time to finish (and you may be quicker than I am!).

The polar graph is a circle with radius 1/2 centered at (0,1/2), i.e.

x² +(y-.5)² = 1/4

The derivative by implicit differentiation is -x/(y-1/2).

Now you need the Cartesian co-ordinates of the point point with r=.5sqrt(3) and theta=-pi/3

(These are x=-.25sqrt(3) and y=3/4)

When you have that plug it into the expression for the derivative to get the slope of the tangent.

(The slope is .25sqrt(3)/[(3/4)-(1/2)] = sqrt(3)....)

Then use the point-slope formula to get the equation for the tangent line.

(You get y=sqrt(3)(x+.25sqrt(3)+.75....)

I will work on this and get back to you...of course, in the meantime another tutor may help you further.(I have added additional lines to the solution between parentheses and I drew the solution on DESMOS to be sure it is correct.)