# x=5+3costheta , y=-2+sintheta Find all points (if any) of horizontal and vertical tangency to the curve.

x=5+3cos theta

y= -2+sin theta

First, take the derivative of x and y with respect to theta.

dx/(d theta) = -3sin theta

dy/(d theta) = cos theta

Take note that the slope of a tangent is equal to dy/dx.

m= dy/dx

To determine the dy/dx of a parametric equation, apply the formula:

dy/dx= (dy/(d theta))/(dx/(d theta))

When the tangent line is horizontal, the slope is zero.

0= (dy/(d theta))/(dx/(d theta))

This occurs when dy/(d theta)=0 and dx/(d theta) !=0 . So setting the derivative of y equal to zero yields:

dy/(d theta) = 0

cos theta = 0

theta_1= pi/2+2pin

theta_2=(3pi)/2 + 2pin

(where n is any integer)

So the graph of the parametric equation has horizontal tangent at these values of theta.

To determine the points (x,y), plug-in the values of theta to the given parametric equation.

theta_1 =pi/2+2pin

x=5+3cos(pi/2+2pin)=5+3cos(pi/2)=5+3*0=5

y=-2+sin(pi/2+2pin)=-2+sin(pi/2)=-2+1=-1

theta_2 = (3pi)/2+2pin

x=5+3cos((3pi)/2+2pin)=5+3cos((3pi)/2)=5+3*0=5

y=-2+sin((3pi)/2+2pin)=-2+sin((3pi)/2)=-2+(-1)=-3

Therefore, the graph of the parametric equation has horizontal tangent at points (5,-1) and (5,-3).

Moreover, when the tangent line is vertical, the slope is undefined.

u n d e f i n e d= (dy/(d theta))/(dx/(d theta))

This occurs when dx/(d theta)=0  and  dy/(d theta)!=0 . So, setting the derivative of x equal to zero yields:

dx/(d theta) = 0

-3sin theta = 0

sin theta = 0

theta_1 = 2pin

theta_2= pi+2pin

(where n is any integer)

So the graph of the parametric equation has vertical tangent at these values of theta.

To determine the points (x,y), plug-in the values of theta to the given parametric equation.

theta_1=2pin

x=5+3cos(2pin)=5+3cos(2pi) =5+3*1=8

y=-2+sin(2pin)=-2+2sin(2pi)=-2+0=-2

theta_2=pi+2pin

x=5+3cos(pi+2pin)=5+3cos(pi)=5+3(-1)=2

y=-2+sin(pi+2pin)=-2+sin(pi)=-2+0=-2

Therefore, the graph of the parametric equation has vertical tangent at points (8,-2) and (2,-2).