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Olin [163]
3 years ago
7

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto

R, where the sides of S are parallel to the u- and v- axes. (Let u play the role of r and v the role of θ. Enter your answers as a comma-separated list of equations.) R lies between the circles x2 + y2 = 1 and x2 + y2 = 7 in the first quadrant
Mathematics
1 answer:
ruslelena [56]3 years ago
6 0
X(u, v) = (2(v - c) / (d - c) + 1)cos(pi * (u - a) / (2b - 2a))
y(u, v) = (2(v - c) / (d - c) + 1)sin(pi * (u - a) / (2b - 2a))

As v ranges from c to d, 2(v - c) / (d - c) + 1 will range from 1 to 3, which is the perfect range for the radius. As u ranges from a to b, pi * (u - a) / (2b - 2a) will range from 0 to pi/2, which is the perfect range for the angle. So, this maps the rectangle to R.
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Answer:

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Step-by-step explanation:

As we can see in the figure that

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So by this

m(\widehat{RS})=m(\angle ROS)

m(\widehat{ST})=m(\angle SOT)

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m(major arc RQ) = m(∠QOR)

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m(\widehat {RS})+m(\widehat{ST})+m(\widehat{TQ}) + m(major\ arc RQ) = m(\angle ROS)+m(\angle SOT)+m(\angle TOQ)+m(\angle QOR)

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Therefore

m(\widehat {RS})+m(\widehat{ST})+m(\widehat{TQ}) + m(major\ arc RQ) = 360\ degree

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