sketch the curve represented by the parametric equations x=cos2t, y=sint for 0<=t<=360
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We have parametric equations x equals cosine of 2t and y equals sine of t, where t ranges from 0 to 360 degrees. Let's trace this curve by watching how the point moves as t increases from 0 to 360 degrees.
To eliminate the parameter, we use the double angle identity cosine of 2t equals 1 minus 2 sine squared t. Since y equals sine t, we can substitute to get x equals 1 minus 2y squared. Rearranging gives us y squared equals 1 minus x over 2. This is the equation of a parabola that opens to the left with vertex at (1, 0).
Now let's determine the range of x and y values. Since y equals sine t for t from 0 to 360 degrees, y ranges from negative 1 to positive 1. Since x equals cosine of 2t, and 2t ranges from 0 to 720 degrees, x also ranges from negative 1 to positive 1. The curve is restricted to this rectangular region. The key boundary points are: when x equals 1, y equals 0, giving us the vertex at (1, 0). When x equals negative 1, y equals plus or minus 1, giving us the points (-1, 1) and (-1, -1).
Let's verify our analysis by checking key points. At t equals 0 degrees, x equals cosine of 0 which is 1, and y equals sine of 0 which is 0, giving us point (1, 0). At t equals 90 degrees, x equals cosine of 180 which is negative 1, and y equals sine of 90 which is 1, giving us point (-1, 1). At t equals 270 degrees, we get point (-1, -1). These points confirm that our curve is indeed the parabola segment from vertex (1, 0) to the endpoints (-1, 1) and (-1, -1).
In summary, the parametric equations x equals cosine 2t and y equals sine t for t from 0 to 360 degrees represent a segment of the parabola y squared equals 1 minus x over 2. This parabola opens to the left with its vertex at (1, 0). The curve is restricted to the domain negative 1 to positive 1 for x, and range negative 1 to positive 1 for y. The endpoints are at (-1, 1) and (-1, -1), and the curve is symmetric about the x-axis. This completes our sketch of the parametric curve.