Question1. (Section 10.4, Problem 10)
Find the Cartecian equation in terms of $x$ and $y$ $x(t)=\sin t, y(t)=2+\cos 2 t$

Studdy Solution

STEP 1

1. We are given parametric equations $x(t) = \sin t$ and $y(t) = 2 + \cos 2t$ .
2. We need to eliminate the parameter $t$ to find a Cartesian equation relating $x$ and $y$ .

STEP 2

1. Express $t$ in terms of $x$ using the equation $x(t) = \sin t$ .
2. Substitute the expression for $t$ into the equation for $y(t) = 2 + \cos 2t$ .
3. Simplify the resulting expression to find the Cartesian equation.

STEP 3

Express $t$ in terms of $x$ using the equation $x(t) = \sin t$ :
$x = \sin t$
To find $t$ , we take the inverse sine (arcsin) of both sides:
$t = \arcsin(x)$

STEP 4

Substitute the expression for $t$ into the equation for $y(t) = 2 + \cos 2t$ :
First, express $\cos 2t$ in terms of $\sin t$ using the double angle identity for cosine:
$\cos 2t = 1 - 2\sin^2 t$
Since $\sin t = x$ , we have:
$\cos 2t = 1 - 2x^2$
Substitute this into the equation for $y$ :
$y = 2 + \cos 2t = 2 + (1 - 2x^2)$

STEP 5

Simplify the expression for $y$ :
$y = 2 + 1 - 2x^2$ $y = 3 - 2x^2$
The Cartesian equation in terms of $x$ and $y$ is:
$y = 3 - 2x^2$

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