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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

Mathematics
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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?
Diffrentiation with respect to time.. @wio
So do you think you can find those derivatives, and then solve for \(\theta\)?

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Other answers:

\[\sec^2 \theta =8 \cos \theta]
\[\sec^2 \theta =8 \cos \theta\]
Now, what is \(\sec^2(\theta )\) in terms of \(\sin(\theta) \) and \(\cos(\theta)\)?
i know is sec = 1/ cos
Do you still need help solving for \(\theta\)?
yes..
Ok so we have \[\Large \frac{1}{\cos^2(\theta)} = 8\cdot \cos(\theta)\]How can we isolate \(\theta \) further?
what will happen next?? no idea. -_-
How about we multiply both sides by \(\cos^2(\theta)\)? Try that.
then it will become 1= 8 cos^3 theta ??
Yes! So what about getting rid of the coefficient?
1/8 = cos ^3 theta ??
Now it's just algebra. We learned that long ago.
How do you get rid of an exponent?
hmm i dont know can u help about it?
Why don't you take the cubed root of both sides?
oww okay I get it :) THanks
Just remember that you want the smallest positive \(\theta \), and that \(\cos(\theta)\) must also be positive since they should be increasing.
Otherwise there would be many solutions!
I get theta = 60 is that correct?
Yes! http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/300px-Unit_circle_angles_color.svg.png
Thank you :)
Great explanation @wio :)

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