Converting parametric equations into Cartesian equations

This article continues from the previous article. It may be a good idea to read through it before attempting this. This article explores how to convert parametric equations into a Cartesian equation. We shall look at some examples below.

Example

In this example we’re going to convert the following parametric equations into a Cartesian equation.
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Here we’re going to use the trigonometric ration which states that;
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We have to eliminate t and write the equations in terms of y and x. We know that given the parametric equations above the following is true;
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Using the ration above that must mean that;
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This is because;
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You may have noticed that (x – 2)² + (y + 3) ² = 1 is an equation for a circle with centre (a, b) and radius r.

Example

Here is another example. In this example we’re going to find the Cartesian equation of the following parametric equations;
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We have to eliminate t and write the equations in terms of y and x. We can see from above that;
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Remember that sin 2t = 2sintcost that must mean that;
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…we also know from above that x = sin t therefore;
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…we have eliminated the first t.
Again here we can use the trigonometric ratio;
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…using it we can see that;
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But we know that x=sin t. That must mean that;
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…and thus…
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We have already found that;
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…so replacing cos t gives;
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We have managed to write two parametric equations as Cartesian equations.