Expert Answer to "Eliminate the parameter to find a Cartesian equation..." - Plainmath

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aidmoon2x

Answered question

2021-11-17

Eliminate the parameter to find a Cartesian equation of the curve $x = \sin t$ , $y = \csc t$ , $0 < t < \frac{π}{2}$

Answer & Explanation

Parminquale

Beginner2021-11-18Added 17 answers

Step 1
Given that

x = \sin t \to

(1)
Step 2
Also given that

y = \csc t = \frac{1}{\sin t}

By definition

\csc t = \frac{1}{\sin t}

Using Eqn(1), We replace

\sin t

with x in the RHS

y = \frac{1}{x}

Step 3
Given that

t \in (0, \frac{π}{2})

Note that

\sin (0)

and

\sin (\frac{π}{2}) = 1

Therefore,

x \in (0, 1)

And because

y = \frac{1}{x}

We have

y \in (1, \infty)

Answer

y = \frac{1}{x}

Where

x \in (0, 1)

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