Math:)

A person is on the outer edge of a carousel with a radius of 20 feet that is rotating counterclockwise around a point that is centered at the origin. What is the exact value of the position of the rider after the carousel rotates 5pi/12 radians?

a. {5[sqrt(2)-sqrt(6)], 5[sqrt(2)+sqrt(6)]}
b. {5[sqrt(2)+sqrt(6)], 5[-sqrt(2)+sqrt(6)]}
c. {5[sqrt(2)+sqrt(6)], 5[sqrt(2)-sqrt(6)]}
d. {5[-sqrt(2)+sqrt(6)], 5[sqrt(2)+sqrt(6)]}

Can someone explain how to do this?

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  1. I will switch to degrees, sensing that you may be more familiar with those units
    5π/12 radians = 75° = 30° + 45°

    draw a unit circle and consider any point (x,y) on that circle.
    Construct any right-angled triangle with base x and height y and hypotenuse 1
    let the angle at the centre be Ø
    then sinØ = y/1 -----> y = sinØ
    and cosØ = x/1 ----> x = cosØ

    If we make the radius r instead of 1, our point becomes
    (rcosØ, rsinØ) or (20cos75°, 20sin75°)

    so we need both cos75° and sin75°

    recall sin(a+b) = sina cosb + cosa sinb
    sin(75) = sin(30+45) = sin30cos45 + cos30sin45
    = (1/2)(√2/2) + (√3/2)(√2/2) = (√2 + √6)/4

    similarly:
    cos(75) = cos(30+45)
    = cos30cos45 - sin30sin45
    = (√3/2)(√2/2) - (1/2)(√2/2)
    = (√6 - √2)/4

    so your point (x,y)
    = (20(√6-√2/4) , 20((√2 + √6)/4)
    = (5√6 - 5√2 , 5√2 + 5√6)

    looks like d) is our match

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