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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: M-D on March 24, 2013, 12:44:02 pm

Title: what is the implied domain and range arccos(2cosec(x)-3)
Post by: M-D on March 24, 2013, 12:44:02 pm: hi,

what would be the implied domain and range of arccos(2cosec(x)-3)?

i got dom=[pi/3,5pi/6] and ran=[0,pi] is it correct?

thanks
Title: Re: what is the implied domain and range arccos(2cos(x)-3)
Post by: leflyi on March 24, 2013, 01:25:22 pm: arcos has domain of [-1,1] and range [0, pi]

hmm

the implied domain of Cos whilst is [-1,1], and range [0, pi] in order for it to become inverse....

So...
arcos(2cos(x)-3)

would have the domain [-1,1] as there is no horizontal translation or dialation of y
and have the range [-3, 2pi-3]

however,

I could be completely wrong..

So feel free to correct me..
Title: Re: what is the implied domain and range arccos(2cos(x)-3)
Post by: M-D on March 24, 2013, 01:31:22 pm: sorry i made a mistake in the question. the question actually is arccos(2cosec(x)-3).

i'll correct it in my initial post
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: Lasercookie on March 24, 2013, 02:16:54 pm: So for arccos(x)
$-1 \le x \le 1$

For arccos(2cosec(x)-3):
$-1 \le 2cosec(x)-3 \le 1$

$2 \le 2cosec(x) \le 4$

$1 \le cosec(x) \le 2$

cosec(x) is equal to 1 when $x = \frac{\pi}{2} + 2\pi n$ where n is an integer.
When cosec(x) is equal to 2, we'll have two general solutions. cosec(x) = 2 when sin(x) = 1/2, so when $x = \frac{\pi}{6} + 2\pi n$ and sine is also positive in the second quadrant, so also $x = \pi - \frac{\pi}{6} + 2\pi n = \frac{5\pi}{6} + 2\pi n$ , where n is an integer.

So we have $\frac{\pi}{2} + 2\pi n \le x \le \frac{5\pi}{6} + 2\pi n$ and $\frac{\pi}{6} + 2\pi n \le x \le \frac{\pi}{2} + 2\pi n$

If that's a bit confusing, then let n = 0 (just for the purposes of visualising it) $\frac{\pi}{2} \le x \le \frac{5\pi}{6}$ and $\frac{\pi}{6} \le x \le \frac{\pi}{2}$ , which is better stated as, $\frac{\pi}{6} \le x \le \frac{5\pi}{6}$ since $\frac{\pi}{6} < \frac{\pi}{2} < \frac{5\pi}{6}$

Hence: $\frac{\pi}{6} + 2\pi n \le x \le \frac{5\pi}{6} + 2\pi n$ which you could state in whatever form you wanted
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: deleted on March 24, 2013, 05:03:08 pm: Quote from: laserblued on March 24, 2013, 02:16:54 pm
So for arccos(x)
$-1 \le x \le 1$

For arccos(2cosec(x)-3):
$-1 \le 2cosec(x)-3 \le 1$

$2 \le 2cosec(x) \le 4$

$1 \le cosec(x) \le 2$

cosec(x) is equal to 1 when $x = \frac{\pi}{2} + 2\pi n$ where n is an integer.
When cosec(x) is equal to 2, we'll have two general solutions. cosec(x) = 2 when sin(x) = 1/2, so when $x = \frac{\pi}{6} + 2\pi n$ and sine is also positive in the second quadrant, so also $x = \pi - \frac{\pi}{6} + 2\pi n = \frac{5\pi}{6} + 2\pi n$ , where n is an integer.

So we have $\frac{\pi}{2} + 2\pi n \le x \le \frac{5\pi}{6} + 2\pi n$ and $\frac{\pi}{6} + 2\pi n \le x \le \frac{\pi}{2} + 2\pi n$

If that's a bit confusing, then let n = 0 (just for the purposes of visualising it) $\frac{\pi}{2} \le x \le \frac{5\pi}{6}$ and $\frac{\pi}{6} \le x \le \frac{\pi}{2}$ , which is better stated as, $\frac{\pi}{6} \le x \le \frac{5\pi}{6}$ since $\frac{\pi}{6} < \frac{\pi}{2} < \frac{5\pi}{6}$

Hence: $\frac{\pi}{6} + 2\pi n \le x \le \frac{5\pi}{6} + 2\pi n$ which you could state in whatever form you wanted

How'd you type all of those cosec and pi values like that?
Is there a tutorial on the forums?
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: Conic on March 24, 2013, 05:05:36 pm: Quote from: deleted on March 24, 2013, 05:03:08 pm
How'd you type all of those cosec and pi values like that?
Is there a tutorial on the forums?
It's called Latex, and there are a few threads on it in the mathematics sub-forum:
List of LaTeX Resources
LaTeX typeset in Maths boards
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: Lasercookie on March 24, 2013, 05:11:02 pm: Quote from: deleted on March 24, 2013, 05:03:08 pm
How'd you type all of those cosec and pi values like that?
Is there a tutorial on the forums?
Yeah, what Conic said. There's a couple of links in my forum signature too, this is kind of a tutorial: How to use LaTeX for typing up maths on ATARNotes
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: Phy124 on March 24, 2013, 05:53:00 pm: This can also be a helpful tool for those starting to use LaTeX.
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: M-D on March 24, 2013, 10:15:17 pm: just going back to the math question. what would be the implied range? thanks
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: availn on March 24, 2013, 10:23:21 pm: Quote from: M-D on March 24, 2013, 10:15:17 pm
just going back to the math question. what would be the implied range? thanks

It would be what you said it was, [0,π]. This is because arccos(x) is at its minimum and maximum when x = 1 and -1 respectively. 2cosec(x) - 3 can equate to any number between 1 and -1, so the range of the function is its normal implied domain.
Title: Re: what is the implied domain and range arccos(2cosec(x)-3)
Post by: JMoxey on March 28, 2013, 10:51:49 pm: This is the exact question that was on the second Calculus 1 assignment at Melbourne Uni.

Breaking Melbourne Uni policy much?