Question

Please help if you understand trigonometry

Answer 1

We're looking for coefficients a, b, and c such that

f(t) = a sin(bt) + c

where t is time.

b represents the period of the sine wave. If the wheel makes 1 revolution every 1.6 seconds, then it has a period of 1.6 s/rev. 1 complete revolution is a turn of 2π radians, so the period is

1.6 s/rev × (1 rev)/(2π rad) = 4π/5 s/rad

The period of sin(bt) is 2π/b, so it follows that

2π/b = 4π/5   ⇒   b = 5/2 = 2.5

a represents the amplitude. Relative to the center of the wheel, the ant's vertical position would oscillate between -11 and 11 inches; this means the amplitude is a = 11.

c represents a vertical shift of the wave. We want the ant's position relative to the ground, not the wheel's center. So we shift the wave up by a constant c such that the lowest and highest points are 3 and 22 + 3 = 25 in above the ground, respectively. In other words, we pick c so that

-11 + c = 3

11 + c = 25

Either equation tells us that c = 14.

So, the equation for the sine wave is

f(t) = 11 sin(2.5t) + 14