Question

What is the coordinates of the child after 125 seconds?

For the above 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.

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 125 seconds will be,

2π × 1/60 × 125 = 25π/6

 

So, the carousel will be at an angle of π/2 + 25π/6 = 28π/6 after 120 seconds.

 

Consider the sine and cosine function in an unit circle,

x = cosθ

y = sinθ

 

Compute the reference angle of 28π/6 as follows:

28π/6 – 2π = 16π/6

16π/6 – 2π = 4π/6

                    = 2π/3

    π – 2π/3 = π/3

 

So, the values of sine and cosine functions at an angle of 28π/6 will be equal to their values at angle of 2π/3. 

 

The angle is in second quadrant. Compute the values of sine and cosine function as π/3.

For angle π/3, the values will be,

cos(π/3) = 1/2

 sin(π/3) = √3/2

 

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

 

Therefore, the co-ordinates of the child after 125 seconds will be (-1/2, √3/2).

Therefore, the co-ordinates of the child after 125 seconds will be (-1/2, √3/2).