Parametric Curve Of X = Cos ⁡ ( T ) X=\cos(t) X = Cos ( T ) , Y = Sin ⁡ 2 ( K T ) Y=\sin^{2}(kt) Y = Sin 2 ( K T ) - Finding The Cartesian Equation Of The Polynomial For Any Natural K K K .

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Introduction

In mathematics, parametric curves are defined by a set of parametric equations that describe the relationship between the variables of the curve. These equations are often used to model real-world phenomena, such as the motion of objects or the behavior of physical systems. In this article, we will explore the parametric curve defined by the equations $x=\cos(t)$ and $y=\sin^{2}(kt)$, where $k$ is a natural number. We will show that this curve is equivalent to the Cartesian graph of a polynomial equation, and we will derive the equation for any value of $k$.

The Parametric Curve

The parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{2}(kt)$

where $t$ is the parameter, and $k$ is a natural number. To find the Cartesian equation of this curve, we need to eliminate the parameter $t$ and express $y$ in terms of $x$.

Eliminating the Parameter

To eliminate the parameter $t$, we can use the trigonometric identity:

$\cos^{2}(t) + \sin^{2}(t) = 1$

We can rewrite the equation $x = \cos(t)$ as:

$\cos(t) = x$

Then, we can square both sides of the equation to get:

$\cos^{2}(t) = x^{2}$

Now, we can substitute this expression into the equation $\cos^{2}(t) + \sin^{2}(t) = 1$ to get:

$x^{2} + \sin^{2}(t) = 1$

Next, we can substitute the expression $y = \sin^{2}(kt)$ into this equation to get:

$x^{2} + y = 1$

The Cartesian Equation

The Cartesian equation of the parametric curve is:

$x^{2} + y = 1$

This is a quadratic equation in $x$, and it represents a parabola that opens to the left. The value of $y$ is determined by the value of $x$, and the equation is valid for any value of $k$.

Examples

Let's consider some examples to illustrate the behavior of the parametric curve for different values of $k$.

$k=1$

For $k=1$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{2}(t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is a parabola that opens to the left, and it is symmetric about the $y$-axis.

$k=2$

For $k=2$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{4}(2t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is also a parabola that opens to the left, and it is symmetric about the $y$-axis.

$k=3$

For $k=3$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{6}(3t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is also a parabola that opens to the left, and it is symmetric about the $y$-axis.

Conclusion

In this article, we have shown that the parametric curve defined by the equations $x=\cos(t)$ and $y=\sin^{2}(kt)$ is equivalent to the Cartesian graph of a polynomial equation. We have derived the equation for any value of $k$, and we have illustrated the behavior of the curve for different values of $k$. The parametric curve is a parabola that opens to the left, and it is symmetric about the $y$-axis. The value of $y$ is determined by the value of $x$, and the equation is valid for any value of $k$.

References

• [1] "Parametric Curves" by MathWorld
• [2] "Cartesian Equations" by Wolfram MathWorld
• [3] "Trigonometric Identities" by MathOpenRef

Further Reading

• "Parametric Curves and Surfaces" by Springer
• "Cartesian Equations and Curves" by Cambridge University Press
• "Trigonometry and Its Applications" by Pearson Education
  Parametric Curve of $x=\cos(t)$, $y=\sin^{2}(kt)$ - Finding the Cartesian Equation of the Polynomial for any natural $k$ ===========================================================

Q&A

Q: What is a parametric curve?

A: A parametric curve is a set of parametric equations that describe the relationship between the variables of the curve. These equations are often used to model real-world phenomena, such as the motion of objects or the behavior of physical systems.

Q: What is the parametric curve defined by the equations $x=\cos(t)$ and $y=\sin^{2}(kt)$?

A: The parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{2}(kt)$

where $t$ is the parameter, and $k$ is a natural number.

Q: How do we eliminate the parameter $t$ to find the Cartesian equation of the curve?

A: We can use the trigonometric identity:

$\cos^{2}(t) + \sin^{2}(t) = 1$

We can rewrite the equation $x = \cos(t)$ as:

$\cos(t) = x$

Then, we can square both sides of the equation to get:

$\cos^{2}(t) = x^{2}$

Now, we can substitute this expression into the equation $\cos^{2}(t) + \sin^{2}(t) = 1$ to get:

$x^{2} + \sin^{2}(t) = 1$

Next, we can substitute the expression $y = \sin^{2}(kt)$ into this equation to get:

$x^{2} + y = 1$

Q: What is the Cartesian equation of the parametric curve?

A: The Cartesian equation of the parametric curve is:

$x^{2} + y = 1$

This is a quadratic equation in $x$, and it represents a parabola that opens to the left. The value of $y$ is determined by the value of $x$, and the equation is valid for any value of $k$.

Q: What happens when $k=1$?

A: When $k=1$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{2}(t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is a parabola that opens to the left, and it is symmetric about the $y$-axis.

Q: What happens when $k=2$?

A: When $k=2$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{4}(2t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is also a parabola that opens to the left, and it is symmetric about the $y$-axis.

Q: What happens when $k=3$?

A: When $k=3$, the parametric curve is defined by the equations:

$x = \cos(t)$ $y = \sin^{6}(3t)$

The Cartesian equation of this curve is:

$x^{2} + y = 1$

This is a parabola that opens to the left, and it is symmetric about the $y$-axis.

Q: Is the parametric curve a parabola for any value of $k$?

A: Yes, the parametric curve is a parabola for any value of $k$. The value of $y$ is determined by the value of $x$, and the equation is valid for any value of $k$.

Q: Can we use this parametric curve to model real-world phenomena?

A: Yes, this parametric curve can be used to model real-world phenomena, such as the motion of objects or the behavior of physical systems. The parametric curve can be used to describe the relationship between the variables of the system, and it can be used to make predictions about the behavior of the system.

Conclusion

In this article, we have answered some common questions about the parametric curve defined by the equations $x=\cos(t)$ and $y=\sin^{2}(kt)$. We have shown that the parametric curve is equivalent to the Cartesian graph of a polynomial equation, and we have derived the equation for any value of $k$. We have also illustrated the behavior of the curve for different values of $k$, and we have discussed the potential applications of this parametric curve in modeling real-world phenomena.