I don't understand how parameters are eliminated

[Eclair_de_XII]

Homework Statement

Eliminate the parameter and identify the graph of the pair of parametric equations. Determine the domain (the set of x-coordinates) and the range (the set of y-coordinates).

x = sin t, y = -3 sin t - 5

Homework Equations

y = -3x - 5
Domain: (-∞,∞)
Range: (-∞,∞)

The Attempt at a Solution

I've already solved this equation by substitution but I don't quite understand how the domain and the range are supposed to make sense. The domain of a sine function can't be higher than one or lower than negative one, right? Because sin²x + cos²x = 1, I don't believe that it's possible. Therefore, the domain of x cannot be (-∞,∞), and the domain of y cannot be (-∞,∞). Can someone explain to me how eliminating parameters is supposed to work?

 

[SteamKing]

You have got the domain and the range of a function confused. The domain is typically the set of x-values over which the function is defined.

 

[Eclair_de_XII]

The domain is typically the set of x-values over which the function is defined.

Do you mean that the range of a sin(t) function is [-1,1], and the domain (the angle) is [0,2π)? I still don't understand how this range and domain are eliminated.

 

[mfb]

The value of a sine function can be smaller than zero, but you are right that it is bounded.
That makes the given domain for x questionable.

 

[vela]

You have three different functions in this problem: x=f(t), y=g(t), and y=h(x), so when you talk about the domain and range, you need to specify which function you're considering. The problem is asking you to find the domain and range of h.

The domain of h is, as noted in the problem statement, all of the possible values x can take. That set happens to also be the range of f. You seem to have an idea of what this is, but [0,1] isn't correct. You might want to look at a plot of the sine function.

 

[Eclair_de_XII]

You seem to have an idea of what this is, but [0,1] isn't correct

It's [-1,1], isn't it? If that's the case, why isn't the range of y [-2,-8]?

 

[vela]

Why do you think it isn't?

 

[Eclair_de_XII]

It's what the online problem said, that's why.

Maybe it made a mistake?

 

[vela]

Sounds like it, or it wants the answer in a different form than the way you entered it.

 

[Eclair_de_XII]

Okay, thanks.

 

[SammyS]

It's [-1,1], isn't it? If that's the case, why isn't the range of y [-2,-8]?

When you use the interval notation, [a, b], b has to be to the right of a on the number line.

The interval, [-2, -8], is improper.

 

[vela]

Good point. I didn't even notice that.