Question

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=\cos (t), y=\sin (t),(0,2 \pi] $$

Short answer

The curve is a circle centered at the origin with radius 1, oriented counterclockwise.

Step-by-step solution

Identify the Parametric Equations

The given parametric equations are \( x = \cos(t) \) and \( y = \sin(t) \), with \( t \) in the range \((0, 2\pi]\). These equations describe the coordinates of points on the curve as a function of the parameter \( t \).

Eliminate the Parameter

To eliminate the parameter \( t \), use the Pythagorean identity \( \cos^2(t) + \sin^2(t) = 1 \). Substitute \( x = \cos(t) \) and \( y = \sin(t) \) into this identity: \( x^2 + y^2 = 1 \). This reveals that the curve is a circle centered at the origin with a radius of 1.

Determine the Orientation of the Curve

Based on the parameter range \( t \in (0, 2\pi] \), the parameter \( t \) starts just greater than 0 and ends at \( 2\pi \). As \( t \) increases from 0 to \( 2\pi \), \( x = \cos(t) \) starts at 1 (at \( t = 0^+ \)), goes to 0 (at \( t = \pi/2 \)), to -1 (at \( t = \pi \)), back to 0 (at \( 3\pi/2 \)), and returns to 1 (just before \( t = 2\pi \)). Simultaneously, \( y = \sin(t) \) starts at 0, goes to 1, to 0, to -1, and returns to 0. This motion describes a counterclockwise revolution around the circle.

Sketch the Curve

With the information gathered, sketch a circle centered at the origin with a radius of 1. The orientation, based on \( t \) increasing, is counterclockwise starting just after the positive x-axis and ending back at it.

Key concepts

Circle sketch

When dealing with parametric equations, sketching the curve involves interpreting how the parameters relate to one another. For equations like \( x = \cos(t) \) and \( y = \sin(t) \), the first step is to identify that each represents part of the Cartesian coordinate system for a circle. Visualizing this requires transforming the parametric equations into a well-known geometric shape.

Start by realizing that these equations describe a circle's position in the plane. By using the Pythagorean identity \( \cos^2(t) + \sin^2(t) = 1 \), we find \( x^2 + y^2 = 1 \). This equation indicates a circle with a center at the origin \((0, 0)\) and a radius of 1.

To illustrate it:

• Draw a point at the origin of the coordinate system.
• Using a compass or freehand, sketch a circle with a radius extending to 1 unit on the axes.
• Ensure the circle passes through points like (1,0), (0,1), (-1,0), and (0,-1).

This simple geometry trick simplifies complex parametric descriptions to a basic, intuitive graphic.

Curve orientation

The orientation of a curve is about understanding the direction in which the curve is traced as the parameter changes. For our circle described by \( x = \cos(t) \) and \( y = \sin(t) \), the orientation is defined by the interval \((0, 2\pi]\).

This interval tells us how \( t \) progresses from slightly more than 0 to \( 2\pi \). As \( t \) increases:

• At \( t = 0^+ \), the point is at \( (1, 0) \).
• At \( t = \pi/2 \), the point moves to \( (0, 1) \).
• At \( t = \pi \), it reaches \( (-1, 0) \).
• At \( 3\pi/2 \), the point is \( (0, -1) \).
• Just before \( t = 2\pi \), it returns to \( (1, 0) \).

This movement from quadrants shows us the points are traced in a counterclockwise direction. To visualize this, imagine tracing a path along the perimeter where every increase in \( t \) moves you forward around the circle. Consider tracing with a pencil starting at the east position of a clock, moving naturally upward and around until you return to the start.

Pythagorean identity

The Pythagorean identity is a fundamental trigonometric principle: \( \cos^2(t) + \sin^2(t) = 1 \). This identity helps connect the trigonometric functions of sine and cosine, allowing us to see them as components on a unit circle.

For any angle \( t \) on this unit circle:

• The horizontal distance (or x-coordinate) from the origin is \( \cos(t) \).
• The vertical distance (or y-coordinate) is \( \sin(t) \).

By squaring \( \cos(t) \) and \( \sin(t) \), and adding them, we sum to 1, indicating they sit perfectly on the circle's perimeter. This directly supports our earlier conclusion; they form a circle with a radius of 1.

Thus, the identity is not just a calculation tool but also a geometric portrayal of points along a circle. Recognizing the role of this identity simplifies understanding the shape and orientation of curves described by parametric equations, transforming them into tangible, circular paths around a central point.