## anonymous 4 years ago 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

1. anonymous

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?

2. anonymous

Diffrentiation with respect to time.. @wio

3. anonymous

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

4. anonymous

$\sec^2 \theta =8 \cos \theta] 5. anonymous \[\sec^2 \theta =8 \cos \theta$

6. anonymous

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

7. anonymous

i know is sec = 1/ cos

8. anonymous

Do you still need help solving for $$\theta$$?

9. anonymous

yes..

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

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

14. anonymous

Yes! So what about getting rid of the coefficient?

15. anonymous

1/8 = cos ^3 theta ??

16. anonymous

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

17. anonymous

How do you get rid of an exponent?

18. anonymous

hmm i dont know can u help about it?

19. anonymous

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

20. anonymous

oww okay I get it :) THanks

21. anonymous

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

22. anonymous

Otherwise there would be many solutions!

23. anonymous

I get theta = 60 is that correct?

24. anonymous

@wio

25. anonymous
26. anonymous

Thank you :)

27. hartnn

Great explanation @wio :)