anonymous
  • anonymous
Give any values in the itnerval [0,2pi] for which the given function has a horizontal tangent line. y= cosx / cscx
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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tkhunny
  • tkhunny
There are lots of hard ways and one relatively easy way. How's your trigonometry?
anonymous
  • anonymous
Well what I did was I first found the derivative of the function, ended up being -tanx+cotx / csc^2(x) and set that equal to zero
anonymous
  • anonymous
So basically I set -tanx+cotx = 0 but then that's where I have trouble.

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tkhunny
  • tkhunny
Yeah, that's horrible. Let's do something else. csc(x) = 1/sin(x). Substitute that.
anonymous
  • anonymous
So then we get y = cosxsinx , the derivative being y= -sin^2x + cos^2x
tkhunny
  • tkhunny
That is another way to go. \(\sin(2x) = 2\sin(x)\cos(x)\) - Substitute that.
anonymous
  • anonymous
Where are you getting the sin2x from?
tkhunny
  • tkhunny
This is why as asked about your trig skills. This is an important identity. \(\sin(x)\cos(x) = ½(2\sin(x)\cos(x)) = ½\sin(2x)\)
anonymous
  • anonymous
I see. I'm not that advanced into trig yet. The process I'm supposed to be using is finding the derivative of the function and solving the problem by setting the derivative equal to zero.
tkhunny
  • tkhunny
What advanced. Trigonometry is a prerequisite to calculus. You should have these identities will in hand before you ever hear the word "derivative". You also had a chance to see it back here... With \(y = \sin(x)\cos(x)\), you correctly found the derivative \(y' = -\sin^{2}(x) + \cos^{2}(x)\). You absolutely should have enough trig to recognize this as \(y' = \cos(2x)\) Anyway, what do you get for the derivative from \(y = ½\sin(2x)\)?
anonymous
  • anonymous
y'=cos2x
anonymous
  • anonymous
And then the points would be pi/4, 3pi/4, 5pi/4, 7pi/4
tkhunny
  • tkhunny
Isn't it awesome that we got there two different ways?! Note: You could have gotten there the way you started out. Always be on the lookout for ways to improve (simplify) you life. This reduces errors if nothing else. Brush up on your trigonometry!

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