Eliminate the Parameter x=tcos(t) , y=tsin(t) , t=pi

$x = t \cos (t)$ , $y = t \sin (t)$ , $t = π$

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = t \cos (t)$

Rewrite the equation as $t \cos (t) = x$ .

$t \cos (t) = x$

Divide each term by $t$ and simplify.

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Divide each term in $t \cos (t) = x$ by $t$ .

$\frac{t \cos (t)}{t} = \frac{x}{t}$

Cancel the common factor of $t$ .

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Cancel the common factor.

$\frac{t \cos (t)}{t} = \frac{x}{t}$

Divide $\cos (t)$ by $1$ .

$\cos (t) = \frac{x}{t}$

Subtract $\frac{x}{t}$ from both sides of the equation.

$\cos (t) - \frac{x}{t} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, t, 1$

Since $1, t, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $t^{1}$ .

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $t^{1}$ is $t$ itself.

$t^{1} = t$

$t$ occurs $1$ time.

The LCM of $t^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$t$

Multiply each term by $t$ and simplify.

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Multiply each term in $\cos (t) - \frac{x}{t} = 0$ by $t$ in order to remove all the denominators from the equation.

$\cos (t) \cdot t - \frac{x}{t} \cdot t = 0 \cdot t$

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $t$ .

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Move the leading negative in $- \frac{x}{t}$ into the numerator.

$\cos (t) t + \frac{- x}{t} \cdot t = 0 \cdot t$

Cancel the common factor.

$\cos (t) t + \frac{- x}{t} \cdot t = 0 \cdot t$

Rewrite the expression.

$\cos (t) t - x = 0 \cdot t$

Reorder factors in $\cos (t) t - x$ .

$t \cos (t) - x = 0 \cdot t$

Multiply $0$ by $t$ .

$t \cos (t) - x = 0$

Add $x$ to both sides of the equation.

$t \cos (t) = x$

Divide each term by $t$ and simplify.

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Divide each term in $t \cos (t) = x$ by $t$ .

$\frac{t \cos (t)}{t} = \frac{x}{t}$

Cancel the common factor of $t$ .

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Cancel the common factor.

$\frac{t \cos (t)}{t} = \frac{x}{t}$

Divide $\cos (t)$ by $1$ .

$\cos (t) = \frac{x}{t}$

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = (\frac{x}{t}) \sin (\frac{x}{t})$

Eliminate the Parameter x=tcos(t) , y=tsin(t) , t=pi

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