Q: Given the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$ for $$t \in (0, \pi)$$, which of the following is the correct Cartesian equation relating $$x$$ and $$y$$?

$$y = \dfrac{1}{x}$$,\; x \neq 0$

Question 1

Given the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$ for $$t \in (0, \pi)$$, which of the following is the correct Cartesian equation relating $$x$$ and $$y$$?

Accepted Answer

$$y = \dfrac{1}{x}$$,\; x 
eq 0$

Question 2

For the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$, what is the domain of $$x$$?

Accepted Answer

$$x \in [-1, 1],\; x 
eq 0$$

Question 3

If $$x = \sin(t)$$ and $$y = \csc(t)$$ for $$t \in (0, \pi)$$, what is the range of $$y$$?

Accepted Answer

$$y \in (-\infty, -1) \cup (1, \infty)$$

Question 4

Suppose you are graphing the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$ for $$t \in (0, \pi). $$Which of the following points is NOT on the curve?

Accepted Answer

$$(0, 0)$$

Question 5

Given $$x = \sin(t)$$ and $$y = \csc(t)$$, if $$t = \dfrac{\pi}{6}$$, what is the corresponding $$(x, y)$$ point?

Accepted Answer

$$\left(\dfrac{1}{2}, 2\right)$$

Question 6

For the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$, what happens to $$y as$$ $$t$$ approaches $$0$$ from the right?

Accepted Answer

$$y$$ approaches $$\infty$$

Question 7

If $$x = \sin(t)$$ and $$y = \csc(t)$$, for which value of $$t in$$ $$(0, \pi)$$ does $$x = 1$$?

Accepted Answer

$$t = \dfrac{\pi}{2}$$

Question 8

Given $$x = \sin(t)$$ and $$y = \csc(t)$$, which of the following statements is TRUE?

Accepted Answer

The graph is undefined at $$x = 0.$$

Question 9

If $$x = \sin(t)$$ and $$y = \csc(t)$$, what is the value of $$y$$ when $$x = -\dfrac{1}{2}$$?

Accepted Answer

$$-2$$

Question 10

For the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$, which of the following best describes the behavior of the graph as $$t$$ approaches $$\pi$$ from the left?

Accepted Answer

$$y$$ approaches $$-\infty$$

Question 11

Suppose you want to eliminate the parameter $$t$$ from $$x = \sin(t)$$ and $$y = \csc(t). $$Which substitution is most useful?

Accepted Answer

Substitute $$t = \arcsin(x)$$ into $$y$$

Question 12

Given $$x = \sin(t)$$ and $$y = \csc(t)$$, which of the following is NOT a possible value for $$(x, y)$$?

Accepted Answer

$$(0, 0)$$

Question 13

If $$x = \sin(t)$$ and $$y = \csc(t)$$, what is the slope $$\dfrac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$-\cot(t) \csc(t)$$

Question 14

For the parametric equations $$x = \sin(t)$$ and $$y = \csc(t)$$, what is the geometric shape of the graph in the $$xy-$$plane?

Accepted Answer

A hyperbola

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