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    the question is:

    Solve sinx=cos(x+30) for values of x 0<x<360.

    So far I've done:

    sin x = cosxcos30 - sinxsin30

    Now I'm not sure what to do can anyone help?
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    cos30 and sin30 are just numbers (constants).

    Factorise the sinx and use a C2 identity.
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    In other words, you have

    sin(x)=cos(x+30)
    <=> sin(x)=cos(x)cos(30)-sin(x)sin(30)
    <=> sin(x) = [sqrt3cos(x)/2] - [sin(x)/2]
    <=> 3sin(x)/2 = sqrt 3 cos(x) / 2
    <=> 3 sin(x)=sqrt3 cos(x)
    <=> tan(x)= (1/sqrt3)
    <=> x = pi/6 [pi]
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    (Original post by paronomase)
    <=> tan(x)= (1/sqrt3)
    <=> x = pi/6 [pi]
    Don't use implication signs, especially when using them wrong.
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    In formal logic, "<=>" is not implication, but equivalence. It is possible that I've made an error, but where would that be?

    The implication sign is "=>". You may want to check this http://en.wikipedia.org/wiki/Logical_implication
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    (Original post by paronomase)
    In other words, you have

    <=> 3 sin(x)=sqrt3 cos(x)
    <=> tan(x)= (1/sqrt3)
    <=> x = pi/6 [pi]
    How do you get from the first to the second line? Thanks for the help
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    3 sin(x) = sqrt 3 cos (x)
    <=> 3 sin(x) / cos (x) = sqrt 3 (for x different of pi/2 [pi] otherwise you cannot divise by 0)
    <=> 3 tan (x) = sqrt 3 since tan(x) = sin(x) /cos(x)
    <=> tan (x) = sqrt 3 / 3
    <=> tan (x) = 1 / sqrt 3
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    (Original post by meiming8)
    the question is:

    Solve sinx=cos(x+30) for values of x 0<x<360.

    So far I've done:

    sin x = cosxcos30 - sinxsin30

    Now I'm not sure what to do can anyone help?
    sinx = cosxcos30 - sinxsin30
    sinx = sqrt(3)cosx / 2 - sinx / 2
    sinx = sqrt(3)cosx-sinx / 2
    2sinx = sqrt(3)cosx - sinx
    3sinx = sqrt(3)cosx
    3tanx = sqrt(3)
    tanx = sqrt(3)/3
    tanx = sqrt(3)*sqrt(3) / 3*sqrt(3)
    tanx = 3 / 3sqrt(3)
    tanx = 1/sqrt3
    x = 30 or 210
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    (Original post by innerhollow)
    sinx = cosxcos30 - sinxsin30
    sinx = sqrt(3)cosx / 2 - sinx / 2
    sinx = sqrt(3)cosx-sinx / 2
    2sinx = sqrt(3)cosx - sinx
    3sinx = sqrt(3)cosx
    3tanx = sqrt(3)
    tanx = sqrt(3)/3
    tanx = sqrt(3)*sqrt(3) / 3*sqrt(3)
    tanx = 3 / 3sqrt(3)
    tanx = 1/sqrt3
    x = 30 or 210

    x=30 or x=210 and x=pi/6 [pi] is exactly the same thing, in the case where 0<x<2pi...(it should be three bars on the "=" sign though, but I did not want to use latex for one symbol)
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    (Original post by paronomase)
    In formal logic, "<=>" is not implication, but equivalence. It is possible that I've made an error, but where would that be?

    The implication sign is "=>". You may want to check this http://en.wikipedia.org/wiki/Logical_implication
    It is an implication; two implications.


    <==> means ==> and <== where the former means "P implies Q" and the latter means "P is implied by Q". The short form of writing ==> and <== is <==>, so it's an implication meaning "if and only if".
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    (Original post by paronomase)
    x=30 or x=210 and x=pi/6 [pi] is exactly the same thing, in the case where 0<x<2pi...(it should be three bars on the "=" sign though, but I did not want to use latex for one symbol)
    Oh you want it in radians? But the domain is 0^{\circ} \leq x \leq 360^{\circ} you know

    \tan x = 1/\sqrt3
    x = \pi/6\ or\ \pi + \pi/6
    x = \pi/6\ or\ 7\pi/6

    Does anyone know how to enter line breaks into latex without using lots of tags by the way? If you only use one latex tag, it puts everything on the one line.


    (Original post by Swayum)
    It is an implication; two implications.


    <==> means ==> and <== where the former means "P implies Q" and the latter means "P is implied by Q". The short form of writing ==> and <== is <==>, so it's an implication meaning "if and only if".
    Why are you being so fussy and annoying? It's even worse because you're actually wrong. <===> is a two-way implication, as in either one implies the other. For example x=30 <===> sinx=0.5 You can infer either statement from either one.

    ===> is only a one-way implication.
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    (Original post by Swayum)
    It is an implication; two implications.


    <==> means ==> and <== where the former means "P implies Q" and the latter means "P is implied by Q". The short form of writing ==> and <== is <==>, so it's an implication meaning "if and only if".
    You are basically contradicting yourself...
    "<=>" is "<=" AND "=>" so it is TWO implications so it cannot be ONE. Obvious no?
    If P <=> Q, saying that P => Q is only a partial affirmation, and therefore you are wrong (unless you do not want to completely describe the relations between the two expressions)

    Therefore, what is wrong with using " <=> " in my case?
    I will not copy everything again, but what is the mistake in writing:

    sin(x)=cos(x+30) <=> sin(x)= cos(x)cos(30)-sin(x)sin(30)

    Indeed, cos (x+30 ) = cos(x)cos(30) - sin(x)sin(30)

    To rewrite it more simply,

    If you have A=B, and
    [1] A = C => B = C
    [2] B = C => A = C

    Then, with [1] and [2], you can write A=C <=> B=C

    Just replace A by sin (x) and C by cos (x+30) and B by .... and you have the result
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    (Original post by innerhollow)
    Oh you want it in radians? But the domain is 0^{\circ} \leq x \leq 360^{\circ} you know

    \tan x = 1/\sqrt3
    x = \pi/6\ or\ \pi + \pi/6
    x = \pi/6\ or\ 7\pi/6
    Yeah basically I just wrote it in radians and using modular arithmetics because it is easier. Have you done it yet? To explain a little bit...
    I wrote x=pi/6 [pi] (read x is congruent to pi/6 modulus pi)

    This means that x= pi/6 + kpi, where k is an integer
    I precised that x belongs to [0,2pi] therefore k=1 or k=0

    Therefore, x = pi/6 or x=pi/6 + pi = 7pi/6

    Another example: if you want to solve cos (x) = 1/2,you can write either
    (1) x = 60° or x=300°
    (2) x=pi/3 or x=2pi -pi/3 = 5pi/3
    (3) x= pi/3 [2pi] or x=5pi/3 [2pi]

    The advantage of (3) is that it tells you that basically there is an infinite number of solutions, if you add 2pi each time.
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    (Original post by paronomase)
    Yeah basically I just wrote it in radians and using modular arithmetics because it is easier. Have you done it yet? To explain a little bit...
    I wrote x=pi/6 [pi] (read x is congruent to pi/6 modulus pi)

    This means that x= pi/6 + kpi, where k is an integer
    I precised that x belongs to [0,2pi] therefore k=1 or k=0

    Therefore, x = pi/6 or x=pi/6 + pi = 7pi/6

    Another example: if you want to solve cos (x) = 1/2,you can write either
    (1) x = 60° or x=300°
    (2) x=pi/3 or x=2pi -pi/3 = 5pi/3
    (3) x= pi/3 [2pi] or x=5pi/3 [2pi]

    The advantage of (3) is that it tells you that basically there is an infinite number of solutions, if you add 2pi each time.
    Oh I see. I wondered why you kept writing a pi in square brackets after.

    But wait? Surely that [pi] thing only works for tan, because for sin/cos there is no constant you can add to generate the next possible solution ad infinitum. For example, the first line is what you said.

    \cos x = \frac{1}{2}
    x_1 = \pi/3 [2\pi] ...Add 2pi
    x_2 = 7\pi/3 ...Add 2pi
    x_3 = 11\pi/3

    But it doesn't work, because you're missing out solutions :confused:
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    (Original post by innerhollow)
    Oh I see. I wondered why you kept writing a pi in square brackets after.

    But wait? Surely that [pi] thing only works for tan, because for sin/cos there is no constant you can add to generate the next possible solution ad infinitum. For example, the first line is what you said.

    \cos x = \frac{1}{2}
    x_1 = \pi/3 [2\pi] ...Add 2pi
    x_2 = 7\pi/3 ...Add 2pi
    x_3 = 11\pi/3

    But it doesn't work, because you're missing out solutions :confused:
    What solutions am I missing? Is it because I divided by cos x? :confused:
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    Yes, it does not always work

    If you are looking at an interval [0,2pi] then if you have an equation of the form cos(x)=k or sin(x)=k you solve it and you get X1 = ... and X2 = ...
    Here, you cannot add a constant, as you said.

    For cos(x)=k, you have two solutions: one that is X1 and the other that is X2=-X1 (for example cos(x)=1/2 <=> x=pi/3 or x=-pi/3

    For sin(x)=k, you have two solutions X1 and X2=pi-X1
    (for example, sin(x) = 1/2 <=> x= pi/6 or x= pi-pi/6=5pi/6)

    Therefore, when solving trigonometric equations, one should be careful.

    Modular arithmetics are really useful, but in trigonometry, the main use is just to show that there is an infinity of solutions for trig equations.

    Here is another example: you have to solve cos(x)sin(x)=0
    one way to do it is to write sin(x)cos(x)=0 <=> sin(x)=0 or cos(x)=0
    <=> x=0, or x=pi/2 or x=pi or x=3pi/2 or x=2pi

    A simplier way to write the solutions is just x=0[pi/2]
    This means x=kpi/2, k being an integer and therefore you have ALL the solutions in one concise form.

    However, I have to admit that this notation is more used in arithmetics problems, and that using them in trigonometry is just "laziness" in a way.
 
 
 
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