Results 1 to 6 of 6

Math Help - Figuring out the number of possible answers in an interval

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    6

    Figuring out the number of possible answers in an interval

    I was recently exposed to the following problem on my homework and I am stumped:

    log(base 5)((sin(x))=-1/2

    I have no idea how to proceed. This was my thought process working out the problem:

    log(base b)(x) = y

    b^y = x

    Therefore:

    5^(-1/2) = sin x

    - sqrt(5) = sin x

    .... but I am at a loss to try and derive the number of possible answers from this answer. The interval is [0, 2pi). Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by DaGr8Gatzby View Post
    I was recently exposed to the following problem on my homework and I am stumped:

    log(base 5)((sin(x))=-1/2

    I have no idea how to proceed. This was my thought process working out the problem:

    log(base b)(x) = y

    b^y = x

    Therefore:

    5^(-1/2) = sin x

    -sqrt(5) = sin x

    \textcolor{red}{5^{-\frac{1}{2}} = \frac{1}{\sqrt{5}} = \sin{x}}

    .... but I am at a loss to try and derive the number of possible answers from this answer. The interval is [0, 2pi). Thank you.
    two solutions ... one in quad I and one in quad II

    ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    6
    Ok, say I had to do this without a calculator, what would be the indicator that there are 2 solutions here?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,689
    Thanks
    617
    Hello, DaGr8Gatzby!

    You said you have no idea, but you did quite good!


    Solve for x\!:\;\;\log_5(\sin x)\:=\:\text{-}\tfrac{1}{2} .for 0 \leq x < 2\pi

    We have: . \log_5(\sin x) \:=\:\text{-}\tfrac{1}{2} \quad\Rightarrow\quad \sin x \:=\:5^{-\frac{1}{2}} \:=\:\frac{1}{\sqrt{5}}

    Therefore: . x \;=\;\arcsin\left(\frac{1}{\sqrt{5}}\right) \;\approx\;\begin{Bmatrix}0.464 \\ 2.678 \end{Bmatrix}

    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by DaGr8Gatzby View Post
    Ok, say I had to do this without a calculator, what would be the indicator that there are 2 solutions here?
    the values of sin(x) increase from 0 to 1 when x goes from 0 to pi/2, and decrease from 1 to 0 when x goes from pi/2 to pi.

    since 0 < \frac{1}{\sqrt{5}} < 1 , there are two solutions in the quadrants stated earlier.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Nov 2009
    Posts
    6
    Thanks guys. This helped me clear things up tremendously. I have other problems and will post more Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: June 20th 2011, 07:10 AM
  2. interval graphs and chromatic number
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 22nd 2010, 01:52 PM
  3. Replies: 5
    Last Post: August 27th 2009, 09:27 AM
  4. Replies: 5
    Last Post: April 30th 2008, 06:17 AM
  5. Prime Number on an interval
    Posted in the Number Theory Forum
    Replies: 9
    Last Post: September 30th 2006, 06:35 PM

Search Tags


/mathhelpforum @mathhelpforum