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September 23rd, 2008, 12:32 AM   #1
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find the exact value of the following expression help

ok, ive got this problem, and all i got was a wrong on it, so im needing some help...

"if cot theta = 3/4, find the exact value of the following expression:

cot(theta+pi)-cot(theta-3pi)+tan(-pi)+cot(theta+5pi)"

my work went like this...

cot(3/4 + pi) - cot (3/4 - 3pi)+ tan(-3/4) + cot(3/4 + 5 pi)

cot pi 2pi

cot 3pi = -1/0


That was my original work, and im trying the problem out again, so any help would be greatly appreciated
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September 23rd, 2008, 06:16 AM   #2
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Re: find the exact value of the following expression help

You've said that cot(theta)= 3/4 implies cot(theta +pi) = cot(3/4 + pi).
Try that with f(x) = x/4: f(x) = 3/4 -> f(x) = (3/4)/4 = (3/16)/4 = .... So that can't work

Start from the definition of cot: cot(t)=1/tan(t) = 1/(sin(t)/cos(t)) = cos(t)/sin(t)

So, let's plug that in to our equation:
cos(t+?)/sin(t+?)-cos(t-3?)/sin(t-3?)+sin(-?)/cos(-?)+cos(t+5?)/sin(t+5?) [notice there's a tangent there, so sin(t) and cos(t) are switched]

So, we know that 2? doesn't change the value of sin or cos, right? because we go once around the circle, so we can say:

= cos(t+?)/sin(t+?)-cos(t-?-2?)/sin(t-?-2?)+sin(-?)/cos(-?)+cos(t+?+4?)/sin(t+?+4?) [factoring out the multiples of 2? from inside each trig function]
= cos(t+?)/sin(t+?)-cos(t-?)/sin(t-?)+sin(-?)/cos(-?)+cos(t+?)/sin(t+?) [removing those multiples 2?]

Now, going halfway around the circle clockwise (-?) gets us to the same place as halfway around the circle counterclockwise(?), so t+?= t-?
= cos(t+?)/sin(t+?)-cos(t+?)/sin(t+?)+sin(?)/cos(?)+cos(t+?)/sin(t+?) [making all the trigfunction(-?) into trigfunction(?)]
= sin(?)/cos(?)+cos(t+?)/sin(t+?) [the first two terms cancel]
= 0/1 + cos(t+?)/sin(t+?) [sin(?)=0, cos(?)=1]
= cos(t+?)/sin(t+?)

Think about what going halfway around the circle does to sin(t) and cos(t). If you're above the x axis (sin is positive), it takes you below (sin is negative) just as far as you were above, and vice versa. If you're to the left of the y axis (cos is negative), it'll take you right of the y axis (cos is positive) just as far. So either way, adding ? to the angle will reverse the sign, so we have:
= -cos(t)/(-sin(t))
= cos(t)/sin(t) [negatives cancel]
= cot(t)
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September 23rd, 2008, 06:35 AM   #3
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Re: find the exact value of the following expression help

cknapp has done a marvelous job here, but the underlying problem is much deeper. There is a total lack of understanding of underlying principles by Tcanuth, and that will not be overcome by rewriting the text here. He really does need to spend time with a tutor to go over each of these areas very carefully, with finite examples of each one, and, more importantly, one-on-one discussion of each one and its connections with other similar functions. The final cot pi 2pi just lost me entirely. There is no methodology there at all. I'm reminded of the one class who were having difficulty in algebra. When I suggested that all they saw was x's and y's all over the page, they all nodded [they had already "passed" a course in algebra the previous year ...with another teacher.]

So, my best advice here is to get a tutor and go over it all VERY carefully. I tutor now, and the last one, a young lady, getting zeros on her tests improved, and finished the semester with 68%. This year she just did her first test and the mother informed me she had obtained 80%. She worked very hard for that, but you can't get the smile off her face. Tutors help.

P.S. I tutor for cancer. That is, parents write a tax-deductible cheque for cancer research. I'm not looking for pupils, just suggesting that people here might choose to do the same.
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September 23rd, 2008, 07:43 AM   #4
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Re: find the exact value of the following expression help

Quote:
Originally Posted by Dave
The final cot pi 2pi just lost me entirely.
I agree wholeheartedly with your post, Dave, but I can see where that's coming from. it's the same logic that led from cot(theta)=3/4 to cot(3/4+pi)

What follows is wrong
If you let cot be a constant, and you miss the tan in the equation
cot(3/4 + pi) - cot (3/4 - 3pi)+ tan(-3/4) + cot(3/4 + 5 pi)
becomes
cot(3/4 + pi) - cot (3/4 - 3pi)+ cot(-3/4) + cot(3/4 + 5 pi), and factoring out the "cot" gives his solution (except there's a negative that wasn't carried through)
*****

The problem is a lack of understanding of functions, specifically trig functions. This is easier to rectify than a complete failure to grasp algebra. The brain is working, but not all the pieces are there.
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September 23rd, 2008, 11:11 AM   #5
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Re: find the exact value of the following expression help

Quote:
Originally Posted by cknapp
The problem is a lack of understanding of functions, specifically trig functions. This is easier to rectify than a complete failure to grasp algebra. The brain is working, but not all the pieces are there.
I've seen that and more. We use to joke that students would make up their own rule, one of which was, "If x occurs on the same page twice, cancel." The problem is that it was no joke. I actually had two girls in their graduating year do just that [they also cheated.] In a trig identity with a function of x on one side and of x on the other, they really did cancel the x's. I forget what they wound up with. I have to admit identifying with Tcanuth's problem and solution though, and sympathise with him/her. I was, and still am, of course, terrible in history. As a young student I used to make up what I thought was a feasible story, or at least the way I thought history ought to have gone. This is no more right than that was. He really should consider a tutor.
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September 24th, 2008, 09:43 PM   #6
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Did the problem include tan(-pi) or tan(theta-pi)?
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September 25th, 2008, 07:02 AM   #7
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Re: find the exact value of the following expression help

What he has written on the top is tan(-?)
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