I need help writing an equation to model a sine wave on a graph. If you could explain how to get the equation, that would be great. I have a quiz coming up and I need to learn how to put these together.
Assume that a satellite launched from Cape Canaveral eventually has an orbit of 2 hours. It reaches 5600 km, its farthest point north of the equator, 45 minutes after the launch and begins its orbit. One hour later, it is 5600 km south of the equator, its farthest point south.
Thanks!
Assume that a satellite launched from Cape Canaveral eventually has an orbit of 2 hours. It reaches 5600 km, its farthest point north of the equator, 45 minutes after the launch and begins its orbit. One hour later, it is 5600 km south of the equator, its farthest point south.
Thanks!
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First decide your units. Let's say x is in hours, y is in km (+ for north, - for south)
A general sine function has this form:
y = A sin(2(pi)x/B + C) + D
"has an orbit of 2 hours" ... that means the period of the function will be 2 (hours). So B=2.
It varies from 5600 to -5600. So it centres at 0, so D=0. An ordinary sine function varies from +1 to -1, so we want to multiply by 5600. So A = 5600.
So far we have
y = 5600 sin((pi)x + C)
All we need to find is C. The furthest point is at x=3/4 (45 minutes is 3/4 of an hour). So we want
5600 sin((pi)(3/4) + C) = 5600
=> sin((pi)(3/4) + C) = 1
(but sin x = 1 when x = pi/2, so...)
=> (pi)(3/4) + C = pi/2
=> C = -pi/4
A general sine function has this form:
y = A sin(2(pi)x/B + C) + D
"has an orbit of 2 hours" ... that means the period of the function will be 2 (hours). So B=2.
It varies from 5600 to -5600. So it centres at 0, so D=0. An ordinary sine function varies from +1 to -1, so we want to multiply by 5600. So A = 5600.
So far we have
y = 5600 sin((pi)x + C)
All we need to find is C. The furthest point is at x=3/4 (45 minutes is 3/4 of an hour). So we want
5600 sin((pi)(3/4) + C) = 5600
=> sin((pi)(3/4) + C) = 1
(but sin x = 1 when x = pi/2, so...)
=> (pi)(3/4) + C = pi/2
=> C = -pi/4