Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

Verified step by step guidance

1. Recall that the parametric equations \(x = \cos t\) and \(y = \sin t\) describe a point on the unit circle for any real number \(t\), because \(\cos^2 t + \sin^2 t = 1\) holds for all \(t\).
2. The parameter \(t\) typically represents the angle measured from the positive x-axis, moving counterclockwise around the circle.
3. Given the interval \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\), identify which portion of the unit circle these values of \(t\) cover. This interval corresponds to angles starting from \(-90^\circ\) to \$90^\circ$.
4. Visualize or sketch the unit circle and mark the points corresponding to \(t = -\frac{\pi}{2}\) and \(t = \frac{\pi}{2}\). Notice that these points lie on the lower and upper parts of the circle, respectively, and the path traced moves along the right half of the circle.
5. Conclude whether the curve described by these parametric equations over the given interval is a semicircle, and specify which semicircle it represents (left or right half).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.

Unit Circle and Trigonometric Functions

The equations x = cos t and y = sin t trace points on the unit circle as t varies. Since cosine and sine represent the x and y coordinates of a point on the circle, these parametric equations describe a circular path with radius 1.

Domain of the Parameter and Curve Segment

The interval for the parameter t determines which part of the curve is traced. For t between −π/2 and π/2, the parametric equations trace only half of the unit circle, specifically the right semicircle, because cosine is non-negative in this range.