Question

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

Answer 1

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?

Answer 2

Diffrentiation with respect to time.. @wio

Answer 3

So do you think you can find those derivatives, and then solve for \(\theta\)?

Answer 4

\[\sec^2 \theta =8 \cos \theta]

Answer 5

\[\sec^2 \theta =8 \cos \theta\]

Answer 6

Now, what is \(\sec^2(\theta )\) in terms of \(\sin(\theta) \) and \(\cos(\theta)\)?

Answer 7

i know is sec = 1/ cos

Answer 8

Do you still need help solving for \(\theta\)?

Answer 9

yes..

Answer 10

Ok so we have \[\Large \frac{1}{\cos^2(\theta)} = 8\cdot \cos(\theta)\]How can we isolate \(\theta \) further?

Answer 11

what will happen next?? no idea. -_-

Answer 12

How about we multiply both sides by \(\cos^2(\theta)\)? Try that.

Answer 13

then it will become 1= 8 cos^3 theta ??

Answer 14

Yes! So what about getting rid of the coefficient?

Answer 15

1/8 = cos ^3 theta ??

Answer 16

Now it's just algebra. We learned that long ago.

Answer 17

How do you get rid of an exponent?

Answer 18

hmm i dont know can u help about it?

Answer 19

Why don't you take the cubed root of both sides?

Answer 20

oww okay I get it :) THanks

Answer 21

Just remember that you want the smallest positive \(\theta \), and that \(\cos(\theta)\) must also be positive since they should be increasing.

Answer 22

Otherwise there would be many solutions!

Answer 23

I get theta = 60 is that correct?

Answer 24

@wio

Answer 25

Yes!
http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/300px-Unit_circle_angles_color.svg.png

Answer 26

Thank you :)

Answer 27

Great explanation @wio :)