Math Problem Statement

Are you sure you look at the grahs and choose the right one?

Which of the following diagrams represents the parametric curve x = -sin t, y = cos t, 0 ≤ t ≤ π?

Solution

To identify which diagram correctly represents the parametric curve given by:

x=sint,y=cost,0tπ,x = -\sin t, \quad y = \cos t, \quad 0 \leq t \leq \pi,

we can analyze the behavior of xx and yy over the interval 0tπ0 \leq t \leq \pi to understand the path of the curve.

  1. When t=0t = 0:

    • x=sin(0)=0x = -\sin(0) = 0
    • y=cos(0)=1y = \cos(0) = 1
    • The curve starts at the point (0,1)(0, 1).
  2. When t=πt = \pi:

    • x=sin(π)=0x = -\sin(\pi) = 0
    • y=cos(π)=1y = \cos(\pi) = -1
    • The curve ends at the point (0,1)(0, -1).
  3. Intermediate values of tt:

    • For tt between 00 and π\pi, sin(t)\sin(t) varies from 00 to 11 (positive values) and then back to 00. Therefore, sin(t)-\sin(t) will range from 00 to 1-1 and back to 00.
    • cos(t)\cos(t) varies from 11 to 1-1.

From this information, we can deduce that the curve traces a semicircle, moving from (0,1)(0, 1) to (0,1)(0, -1), with xx values remaining negative or zero. This matches a left-facing semicircle centered at the origin.

Diagram Choice:

  • The correct option is (B), which shows a left-facing semicircle from (0,1)(0, 1) to (0,1)(0, -1).

Let me know if you'd like more details on any step.


Additional Questions:

  1. How can you determine the direction of a parametric curve?
  2. Why does the value of tt only go up to π\pi here?
  3. What would happen if the range of tt extended beyond π\pi?
  4. Can parametric curves represent other shapes besides circles?
  5. How would the graph change if x=sintx = \sin t instead of x=sintx = -\sin t?

Tip: When analyzing parametric curves, checking endpoint values and the general shape by looking at intermediate values can help quickly identify the curve’s form.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Parametric Equations
Trigonometric Functions
Curve Sketching

Formulas

x = -sin(t)
y = cos(t)
0 ≤ t ≤ π

Theorems

Behavior of Sine and Cosine Functions over an Interval

Suitable Grade Level

Grades 10-12