Pre-Calc Homework Help: Simplify (sec^2 x)(csc x)/(csc^2 x)(sec x)"

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The discussion focuses on simplifying the expression (sec^2 x)(csc x)/(csc^2 x)(sec x). Participants identify that initial cancellations can lead to (sec x)/(csc x), which can be further simplified to 1/tan x. The conversation then shifts to finding the exact value of sin(5pi/12), with suggestions to use the unit circle and trigonometric identities. It is clarified that 1/4 is incorrect, and the correct approach involves using the sum of angles formula to derive the value. Ultimately, the exact value of sin(5pi/12) is determined to be (√2 + √6)/4.
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Simplify the given expression:
2) (sec^2 x)(csc x)/(csc^2 x)(sec x)
 
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What have you tried?
 
well...i've done this:

(tan^2 x + 1)(csc x)/(cot^2 x + 1)(sec x)

but...i don't know if I'm going in the right direction, for my teacher is horrible, and i don't know where to go from here if i am going in the right direction

any help would be greatly appreciated
 
OK: Let's make the EASIEST cancellations first:
If you look at the expression like this:
\frac{sec^{2}(x)csc(x)}{sec(x)csc^{2}(x)}

isn't there a couple of cancellations which immediately spring to your mind?
 
so is it (sec x)/(csc x) ?
 
math_fortress said:
so is it (sec x)/(csc x) ?
Precisely!
Now, knowing the relation between sec and cos and csc and sin, can you simplify even further?
 
well, since sin/cos = tan...then would sec/csc = 1/tan ?

That's all i can think of
 
No, you have:
\frac{\frac{1}{\cos(x)}}{\frac{1}{\sin(x)}}=\frac{1}{\cos(x)}\frac{1}{\frac{1}{\sin(x)}}=\frac{\sin(x)}{\cos(x)}=tan(x)
 
alright...thanks, I'm starting to get it a little better...
 
  • #10
Wait...this baffles me...

26.) Find the exact value of sin 5pi/12

Is there any way to do this logically w/ a calculator or anything?
 
  • #11
Of course there is! (hopefully you meant without a calculator) That's why the angle is given to you in radians, as a rational multiple of \pi.

Draw the unit circle: what coordinate points do certain angles represent? \pi, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} etc.
 
  • #12
but 5pi/12 isn't on my unit circle...the one's you listed are though...i just don't get how exactly you can find 5pi/12 with information of pi/2, etc...
 
  • #13
would 1/4 make sense for the answer since i got the sin of 5pi/6 to equal 1/2?
 
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  • #14
Try to do this using the trig identity
\cos{2a}=1-2\sin^{2}{a}
 
  • #15
What? So you're saying 1/4 isn't right then?
Do I even need to use trig identities for this type of question?

(I'm not arguing with that identity..i'm just very confused :bugeye:)
 
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  • #16
1/4 is incorrect. Sometimes identities are necessary.
 
  • #17
Yeah, sorry if I gave you the wrong idea. 1/4 is incorrect. You are making the assumption that if I halve the angle, I halve the sine. You can see why that would only be true for a linear relationship right? (which sine is not). I think Sirus has the right technique, since the trig identity involves a term with twice the angle and another with just the angle itself. We know how to work with \frac{\pi}{6}, and multiples of it, so find the cosine of the angle \frac{\5pi}{6} and work from there.
 
  • #18
Another way is to convert the radians to degrees and work from there:

sin5pi/12=sin5(180)/12=sin75=sin(30+45)

now you can just use the formula for the sum of angles:
sin (a+b) = cos(b) sin(a) + sin(b) cos(a)
 
  • #19
use the sum formulas of trig:
sin (a+b)=sin(a) cos(b)+cos(a) sin (b)

now, sin 5pi/12=sin (2pi/12 + 3pi/12), so the exact value of sin 5pi/12 is?
 
  • #20
alright...root 2/4 + root 6/4


Thanks for all help
 
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