bii17
If an angle theta increases uniformly, find the smallest positive value of theta for which tan theta increases 8 times as fast as sin theta
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wio
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Would it be something like this?
\[\Large \frac{d}{d\theta}tan(\theta) =8\cdot \frac{d}{d\theta}sin(\theta) \] Where \[\Large \frac{d}{d\theta}tan(\theta) > 0\]
What is the topic of the curriculum?
bii17
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Diffrentiation with respect to time.. @wio
wio
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So do you think you can find those derivatives, and then solve for \(\theta\)?
bii17
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\[\sec^2 \theta =8 \cos \theta]
bii17
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\[\sec^2 \theta =8 \cos \theta\]
wio
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Now, what is \(\sec^2(\theta )\) in terms of \(\sin(\theta) \) and \(\cos(\theta)\)?
bii17
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i know is sec = 1/ cos
wio
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Do you still need help solving for \(\theta\)?
bii17
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yes..
wio
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Ok so we have \[\Large \frac{1}{\cos^2(\theta)} = 8\cdot \cos(\theta)\]How can we isolate \(\theta \) further?
bii17
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what will happen next?? no idea. -_-
wio
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How about we multiply both sides by \(\cos^2(\theta)\)? Try that.
bii17
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then it will become 1= 8 cos^3 theta ??
wio
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Yes! So what about getting rid of the coefficient?
bii17
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1/8 = cos ^3 theta ??
wio
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Now it's just algebra. We learned that long ago.
wio
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How do you get rid of an exponent?
bii17
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hmm i dont know can u help about it?
wio
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Why don't you take the cubed root of both sides?
bii17
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oww okay I get it :) THanks
wio
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Just remember that you want the smallest positive \(\theta \), and that \(\cos(\theta)\) must also be positive since they should be increasing.
wio
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Otherwise there would be many solutions!
bii17
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I get theta = 60 is that correct?
bii17
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@wio
bii17
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Thank you :)
hartnn
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Great explanation @wio :)