Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. x= -7 cos 2t, y= -7 sin 2t, 0 leq t leq pi

Chardonnay Felix 2021-02-21 Answered
Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. x= 7 cos 2t, y= 7 sin 2t, 0  t  π
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Daphne Broadhurst
Answered 2021-02-22 Author has 109 answers

Step 1 Given: The following parametric equations, x= 7 cos(2t), y= 7 sin(2t), 0  t  π.

Step 2 a) To eliminate the parameter to obtain an equation in x and y. We have, The following parametric equations, (1)x= 7cos(2t)
y= 7sin(2t)(2) Adding equations (1) and (2), we get x2 + y2=(7 cos(2t))2 + (7 sin(2t))2
=49 cos2(2t) + 49 sin2(2t)
=49(cos2(2t) + sin2(2t))
=49(1) By using trignometric identity
=49 An equation in x and y is x2 + y2=49. Step 3 b) To describe the curve and indicate the positive orientation. We have, x2 + y2=(7)2. The curve represents a circle at the origin, (0, 0) and radius, r=7. To indicate the positive orientation, we will use the following parametric equations, x= 7 cos(2t), y= 7 sin(2t), 0  t  π. The interval given is 0  t  π.
x= 7 cos(2(0))= 7(cos(0))= 7(1)= 7
y= 7 sin(2(0))= 7(sin(0))= 7(0)=0. Implies the initial point is (7, 0). For t=π,
x= 7 cos(2 π)= 7(1)= 7y= 7 sin(2 π)= 7(0)=0. Implies the end point is (7, 0). The initial point and the end point are equal, that is, (7, 0). Hence, the orientation is positive in anticlockwise direction.

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