Question

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0,1), that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 90 seconds?

Answer 1

Consider the scenario referred in the textbook. So, the child is at an angle of θ = π/2 initially. The carousel revolves counterclockwise. As it will be revolved by an angle of 2π in 1 hour or 60 minutes, the measurement of the angle by which the carousel revolves in 90 seconds will be,

2π × 1/60 × 90 = 3π

 

So, the carousel will be at an angle of π/2 + 3π = 7π/2 after 90 seconds.

 

Consider the sine and cosine function in a unit circle,

x = cosθ

y = sinθ

 

Compute the reference angle of 7π/2 as follows:

7π/2 – 2π = 3π/2

  3π/2 – π = π/2

 

So, the values of sine and cosine functions at an angle of 7π/2 will be equal to their values at angle of 3π/2. The angle is in third quadrant. Compute the values of sine and cosine function as π/2.

 

For angle π/2, the values will be,

cos(π/2) = 0

 sin(π/2) = 1

 

As the angle 3π/2 is in third quadrant, the x-coordinate and y-coordinates of the required point will be negative.

 

Therefore, the co-ordinates of the child after 90 seconds will be (0, -1).

Therefore, the co-ordinates of the child after 90 seconds will be (0, -1).