Question

A particular curve is represented parametrically by x=cost, y=−sint, t∈[0,2π]. (a) As t increases on [0,2π], in which direction is the point (x(t),y(t)) moving? the movement is clockwise (b) What is the corresponding Cartesian equation for this curve (the equation in x and y only)? Cartesian equation: functionsequation editor =0. Note! xyy=x2+1y−x2−1=0y−x2−1 (c) Give the smallest and largest values of y taken by this curve functionsequation editor ≤y≤ functionsequation editor .

Answer 1

Answer:

a) clockwise sense

b)  x² + y² = 1

c) y ∈ [0,1]

Step-by-step explanation:

a) x = cos (t)       y = -sin (t)

for t ∈ [ 0 , 2π ]

for t = 0   point  P  is    x = cos (0)   =  1   y = -sin (0)   y = 0

The point is on the x axis

Now let´s see what happens for t = π/4  

x = cos (π/4)    x = √2/2    and   y = - sin (π/4)    y = - √2/2

Note  π/4 is 45°

The coordinates of the new point P ´(t = π/4) are in the fourth quadrant

that means the point is moving clockwise sense.

b) x = cos(t)

   y = -sin(t)

Squaring on both sides of each equation we get:

x²  = cos²t

y²  = sin²(t)

Adding side by side

x²  +  y² = cos²t  + sin²t

x²  +  y² = 1

Values for y are    y  ( 0 , 1)