You are given the parametric equations x=2cos(θ),y=sin(2θ). (a) List all of the points (x, y) where the tangent line is horizontal. In entering your answer, list the points starting with the smallest value of x. If two or more points share the same value of x, list those points starting with the smallest value of y. If any blanks are unused, type an upper-case "N" in them. Point 1: (x, y)= ( , )

Using derivatives of the parametric curves and the solving a trigonometric equation, it is found that the points (x,y) are:

Given two parametric curves x(t) and y(t), the slope of the tangent line is given by:

The tangent line is horizontal for:

In this problem, since we are working with trigonometric functions sine and cosine, they will never be simultaneously 0, hence, the equation of interest is:

Then:

Hence:

Hence, the points (x,y) are:

A similar problem is given at link

The solutions listed from the smallest to the greatest are:

x:        

y:      -1         1     -1     1

Step-by-step explanation:

The slope of the tangent line at a point of the curve is:

The tangent line is horizontal when . Then:

, for all

, for all

The first four solutions are:

x:        

y:     1        -1        1     -1

The solutions listed from the smallest to the greatest are:

x:        

y:      -1         1     -1     1

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47