Consider a circle, radius $a$, center at the origin. Let $P(x, y)$ be any point on the circle, and let the angle between $PO$ and the x-axis be $\theta$, then $x=a\cos\theta$ and $y=a\sin\theta$.

These equations, which give the coordinates of any point on the curve in terms of $\theta$, are called parametric equations, and $\theta$ is called a parameter.

Find the Cartesian equation of the locus given parametrically by the equations $x=\sin\theta$, $y=\sin2\theta$

PARAMETER AND

chegendungu

$y=\sin2\theta$, but $\sin2\theta=2\sin\theta\cos\theta$, therefore

$y=2\sin\theta\cos\theta$

$\therefore y^2=4\sin^2\theta\cos^2\theta$

Now $x=\sin\theta$, therefore $1-x^2=\cos^2\theta$, and so the Cartesian equation of the locus is $y^2=4x^2(1-x^2)$.

The process of obtaining parametric equations from a given Cartesian equation is not so easy as the reverse, but one method is illustrated in the next example.

Surround your text in `*italics*`

or `**bold**`

, to write a math equation use, for example, `$x^2+2x+1=0$`

or `$$\beta^2-1=0$$`

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