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Math Help - solving for trig x

  1. #1
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    solving for trig x

    1. sin(x) + sqrt2 = - sinx

    2. 3 tan(x/2)+3=0

    3. solve for cosx ; cos x =1/2
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  2. #2
    Member TheMasterMind's Avatar
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    Quote Originally Posted by william View Post
    1. sin(x) + sqrt2 = - sinx

    2. 3 tan(x/2)+3=0

    3. solve for cosx ; cos x =1/2
    1. isolate sine
    sin(x)+sqrt2=-sinx
    2sin(x)=-sqrt2
    sin(x)=-sqrt2/2

    use the inverse sine function

    sin(x)=-sqrt2/2
    sin^-1(sin(x))=sin^-1(-sqrt2/2)
    x=sin^-1(-sqrt2/2)
    x=-0.7853

    reference angle must be x=0.7853

    sine is negative in quadrant 3 & 4

    now take it from here
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  3. #3
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    Solving for x

    Hello william

    Quote Originally Posted by william View Post
    1. sin(x) + sqrt2 = - sinx

    2. 3 tan(x/2)+3=0

    3. solve for cosx;
    Quote Originally Posted by william View Post
    cos x =1/2


    1

    Add
    \sin x to both sides, and subtract \sqrt{2} from both sides:

    2\sin x = - \sqrt{2}

    \implies \sin x = -\frac{\sqrt{2}}{2}

    Now I'm not sure what sort of answer is needed now. Do you use a calculator to work out a single answer for
    x, giving your answer in degrees? Or do you want lots of answers in radians, involving things like \frac{\pi}{4} and 2n\pi?

    I don't know. But in degrees, possible answers include
    -45^o, 225^o, and so on.

    2

    3 \tan \left(\frac{x}{2}\right) + 3 = 0

    \implies   3 \tan \left(\frac{x}{2}\right)=-3

    \implies \tan \left(\frac{x}{2}\right)=-1

    Again, you have lots of possibilities now. One answer is that

    \frac{x}{2} = -45^o

    So
    x = -90^o

    There are lots of other possibilities...

    3

    By now, I'm guessing that you want all the possible values of
    x if \cos x = \frac{1}{2}

    I reckon the easiest way is to look at a sketch graph of
    y = \cos x, and work them out from that.

    The first positive value is
    60^o (or \frac{\pi}{3} radians), but there's another on the other side of the y-axis, where x = -60^o ( -\frac{\pi}{3})

    The graph repeats every
    360^o, or 2\pi radians, so you get more answers by adding 360^o ( 2\pi radians) on to these two answers. So in degrees:

    -60^o, 60^o, 300^o, 360^o, 660^o, 780^o, ...

    I'll leave it to you to find these answers in radians, if you need to.

    Grandad

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  4. #4
    Member TheMasterMind's Avatar
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    Quote Originally Posted by william View Post
    1. sin(x) + sqrt2 = - sinx

    2. 3 tan(x/2)+3=0

    3. solve for cosx ; cos x =1/2
    2.
    3 tan(x/2)+3=0
    tan(x/2)=-1

    -pi=-pi<x<pi=pi
    tan^-1(tan(x/2))=tan^-1(-1)
    x=2tan^-1(-1)=-1.5707963

    therefore the exact solutions are x=2tab^-1(-1)plusminus2npi

    substitute -1.5707963 to check

    3tan\frac{x}{2}+3=3\tan\frac{-1.5707963}{2}+3=0
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