The pair of parametric equations represent a line, parabola, circle, ellipse, or hyperbola. Name the type of basic curve that the pair of equations represents.
[tex]
\begin{array}{l}
x=3 \sin (5 t) \\
y=3 \cos (5 t)
\end{array}
[/tex]
A. line
B. parabola
C. circle
D. ellipse
E. hyperbola
Answer (2)
Square both parametric equations.
Add the squared equations to eliminate the parameter t .
Apply the trigonometric identity sin 2 ( θ ) + cos 2 ( θ ) = 1 .
Recognize the resulting equation as a circle: c i rc l e .
Explanation
Analyze the parametric equations. We are given the parametric equations:
x = 3 sin ( 5 t )
y = 3 cos ( 5 t )
Our goal is to identify the type of curve these equations represent.
Square both equations. To eliminate the parameter t , we can square both equations:
x 2 = ( 3 sin ( 5 t ) ) 2 = 9 sin 2 ( 5 t ) y 2 = ( 3 cos ( 5 t ) ) 2 = 9 cos 2 ( 5 t )
Add the squared equations. Now, add the squared equations:
x 2 + y 2 = 9 sin 2 ( 5 t ) + 9 cos 2 ( 5 t )
Factor out 9. Factor out the 9:
x 2 + y 2 = 9 ( sin 2 ( 5 t ) + cos 2 ( 5 t ))
Apply the trigonometric identity. Using the trigonometric identity sin 2 ( θ ) + cos 2 ( θ ) = 1 , we have:
x 2 + y 2 = 9 ( 1 ) = 9
Identify the curve. The equation x 2 + y 2 = 9 represents a circle centered at the origin (0, 0) with a radius of 9 = 3 .
Conclusion. Therefore, the parametric equations represent a circle.
Examples
Parametric equations are useful in physics to describe the trajectory of a projectile. For example, if an object is thrown with an initial velocity and angle, its position (x, y) at any time t can be described using parametric equations. Similarly, in computer graphics, parametric equations are used to draw curves and surfaces. For instance, to draw a circle, we can use the parametric equations x = r cos ( t ) and y = r sin ( t ) , where r is the radius and t is the parameter that varies from 0 to 2 π .
The given parametric equations represent a circle, as shown by transforming them into the standard form of a circle's equation, which is x 2 + y 2 = 9 . Thus, the correct answer is option C: circle.
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