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 2 years ago
Given CSC(s)= 13/12, where cos(s)<0 & tan(t)=sqrt3 where 3pi/2 < t < 2pi.
Find exact value of cos(st)
 2 years ago
Given CSC(s)= 13/12, where cos(s)<0 & tan(t)=sqrt3 where 3pi/2 < t < 2pi. Find exact value of cos(st)

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lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0@phi or @jim_thompson5910 or @zepdrix can anyone of you explain this to me please. thank you so much

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Hmm, do you happen to know what the answer is? I came up with something, but I'd hate to explain this to you if I'm wrong lol.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0lol its okay, and no i dont have the answer. i have to correct these problems inorder to retake the test. and i have to turn this in tomorrow morning. Do you mind helping me with these problems, so far this is 1 out of 6 problems. and they are all different.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1I doubt I'll have time :c Important exam tomorrow. But just post your questions with the little @ tag and I'll try to take a look if I get a moment. c: As for this problem.... Hmm, lemme see if I'm on the right track here. cos(s) < 0, so we're in the 2nd or 3rd quadrant.\[\csc(s)=\frac{13}{12} \quad \rightarrow \quad \frac{1}{\sin(s)}=\frac{13}{12} \quad \rightarrow \quad \sin(s)=\frac{12}{13}\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Sine positive, cosine negative, so we're in the 2nd quadrant.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1They told us that t lies in the 4th quadrant. And they gave us a special angle for t. \[\tan(t)=\sqrt3\]

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0how was the cosine negative?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1cos(s) < 0 was in the problems description. Less than zero means it's negative :)

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Lots of little details to this problem, it's easy to miss stuff :D heh

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1So hmm, let's try to solve for t.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1If it was in the first quadrant it would be... lemme try to remember.. it's the larger angle, pi/3 i guess. Since we're in the 4th quadrant we gotta think a lil bit.. it's ummm

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1So in the 4th quadrant is it, 5pi/3 maybe?

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0haha im sorry im superly bad at this, im retaking this class next fall but im going to give it my best shot to retake a test and on the final :)

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0i checked by drawing haha

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0by doing it the same way? but wouldnt we get....hm...the same 12 and 13?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1It's not straight across even though it might look that way based on how I drew it. 12/13 is not a special angle. :D I don't think so at least :3

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\large \cos(st)\]So what I did from here is... Apply the Difference Formula for Cosine. Remember the Sum and Difference formulas?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\large \cos(st)=\cos(s)\cos(t)+\sin(s)\sin(t)\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1This problem is a bit of a doozy.. Once you get to this point though, it's not too bad. Just a bunch of plugging in angles. We know what t is, so we can plug that value into cos(t) and sin(t). We already know what sin(s) is. And by finding the missing side of that triangle above, we can easily determine cos(s).

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1This is a pretty rough problem :D I don't remember doing anything like this in trig lol

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0haha, dang...okay hmmm.. okay so i just plug in right? . lol

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1So looking at our triangle we drew on the coordinate plane, can you determine cos(s)?

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0well the cos s. would be the path. the. that equaled 5, right?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Cosine would be Adjacent over Hypotenuse, soh cah toa :D CAH! C=A/H

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1If this had been the UNIT circle, then yes you would be correct :)

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0lol ya, so 5 right? wait.....lol hold on

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Look at the triangle. The Hypotenuse is ummm 13 I think.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1woops you calculated sin again :O

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.112 is the side OPPOSITE s.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0okay i got 13/12 = 1.0833

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.15 is the side ADJACENT to s. 13 is the HYPOTENUSE :D silly sis ^^

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Keep in mind we're going to leave it as a FRACTION, they want an EXACT answer.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1sin(s)=12/13, we'll be leaving it that way. We want to find cos(s), it should be similar, but it will relate 2 other sides together. not the 12 and 13.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\large \sin(s)=\frac{12}{13}, \quad \cos(s)=\frac{5}{13}\] \[\large \cos(t) \quad \rightarrow \quad \cos\left(\frac{5\pi}{3}\right)=?\]\[\large \sin(t) \quad \rightarrow \quad \sin\left(\frac{5\pi}{3}\right)=?\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Ah crap I just noticed, the cosine should be a NEGATIVE value, remember why?

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0because its in the 3rd quad?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1lol ya c: because of that.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1now you have to think back to your special angles. What do you get for sin(t) and cos(t)? :o

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0crud, i forgot the special angles :/ okay um...sin t would be....  1/2 ?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Ummm let's think. This is an angle where the y value is larger. So the sine is going to be the bigger one, I think it's sqrt3/2.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0oh man :? im so bad at this haha your right

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0the cos is going to be 1/2 then

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1K sounds good :D positive right? since we're in the 4th.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0yes. but how did we get from 2nd to fourth? haha i just noticed that

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1So your problem should give you something like this,\[\large \cos(st)=\cos(s)\cos(t)+\sin(s)\sin(t)\] \[\large \rightarrow \quad \left(\frac{5}{13}\right)\cdot\left(\frac{1}{2}\right)+\left(\frac{12}{13}\right)\cdot\left(\frac{\sqrt3}{2}\right)\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Oh S was in 2nd :) t was in 4th

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1They're making us jump around and use 2 different angles.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0WAIT! i see it, tan t is  sqrt 3, so that means sin/+ cos haha okay 4th Q

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Oh also, they told us that 3pi/2 < t < 2pi These are the boundaries on t, remember where this region lies? :D

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1sqrt3 isn't quite enough information. Because tan t could be +sin/cos right? :) Which is second quad.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0okay so now i solve that equation?? or do we gotta get another anser?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1See the big ugly fractions I plugged in? We just need to simplify that down a tiny bit, NO DECIMAL, and we're finished.

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0haha okay hold on let me try it

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0i got  5/12, haha and i cant do the other one

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1It won't simply very far :) We're just combining fractions. Giving you something like this,\[\huge \cos(st)\quad =\quad \frac{512\sqrt3}{26}\]

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Sounds like you're on the right track though :D

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0dang. haha thank you so much

lilsis76
 2 years ago
Best ResponseYou've already chosen the best response.0just let me know when you have to get off okay :)

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1I'm not positive that's correct. It's a rather burdensome problem. Hopefully we didn't make any silly mistakes in there. :D

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1Post your new questions in NEW threads. That way, just in case I can't get to them, someone else won't have to scroll AAAALL the way down this thread to find your question. :D

phi
 2 years ago
Best ResponseYou've already chosen the best response.1fyi, you got the exact answer \[ \cos(st)= \frac{(5+12\sqrt{3})}{26} \]

phi
 2 years ago
Best ResponseYou've already chosen the best response.1btw, it is helpful to remember that if the lengths of a triangle are 3,,4,5 or 5,12,13 they are right triangles (it saves time solving for the 3rd side if you know this) also, if all 3 legs are multiplied by the same number (example: 10,24,26 (5,12,13 all multiplied by 2), then you still have a right triangle.
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