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bii17

  • 2 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

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  1. wio
    • 2 years ago
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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?

  2. bii17
    • 2 years ago
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    Diffrentiation with respect to time.. @wio

  3. wio
    • 2 years ago
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    So do you think you can find those derivatives, and then solve for \(\theta\)?

  4. bii17
    • 2 years ago
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    \[\sec^2 \theta =8 \cos \theta]

  5. bii17
    • 2 years ago
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    \[\sec^2 \theta =8 \cos \theta\]

  6. wio
    • 2 years ago
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    Now, what is \(\sec^2(\theta )\) in terms of \(\sin(\theta) \) and \(\cos(\theta)\)?

  7. bii17
    • 2 years ago
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    i know is sec = 1/ cos

  8. wio
    • 2 years ago
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    Do you still need help solving for \(\theta\)?

  9. bii17
    • 2 years ago
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    yes..

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

  11. bii17
    • 2 years ago
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    what will happen next?? no idea. -_-

  12. wio
    • 2 years ago
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    How about we multiply both sides by \(\cos^2(\theta)\)? Try that.

  13. bii17
    • 2 years ago
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    then it will become 1= 8 cos^3 theta ??

  14. wio
    • 2 years ago
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    Yes! So what about getting rid of the coefficient?

  15. bii17
    • 2 years ago
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    1/8 = cos ^3 theta ??

  16. wio
    • 2 years ago
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    Now it's just algebra. We learned that long ago.

  17. wio
    • 2 years ago
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    How do you get rid of an exponent?

  18. bii17
    • 2 years ago
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    hmm i dont know can u help about it?

  19. wio
    • 2 years ago
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    Why don't you take the cubed root of both sides?

  20. bii17
    • 2 years ago
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    oww okay I get it :) THanks

  21. wio
    • 2 years ago
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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.

  22. wio
    • 2 years ago
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    Otherwise there would be many solutions!

  23. bii17
    • 2 years ago
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    I get theta = 60 is that correct?

  24. bii17
    • 2 years ago
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    @wio

  25. bii17
    • 2 years ago
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    Thank you :)

  26. hartnn
    • 2 years ago
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    Great explanation @wio :)

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