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sec, cosec and cot question

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    Hi guys
    Wanting to cover some C3 in the summer, and have come across this question which I can not seem to do, was wondering if someone could give me a push in the right direction?

    Simplify cosec(pi/2 - x)
    I've learnt that the next step is writing what cosec x = 1/sin x (How would someone prove this? the book doesn't ...)

    thus :
    cosec(pi/2 - x) = 1/sin (pi/2 - x)

    But what else could I do to simply further?

    P.s Could someone please clarify some of the terminology to me please? I believe an inverse function is when you reflect a function in y=x? so the inverse of y=sin x is y=sin^-1 x ? However Sin^-1 x =/= 1/sin x ? so Cosec is not an inverse function of sin x?
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    (Original post by coolstorybrother)
    cosec x = 1/sin x (How would someone prove this? the book doesn't ...)
    It's the definition - literally, cosec is just a quick way of writing 1/sin

    (Original post by coolstorybrother)
    thus :
    cosec(pi/2 - x) = 1/sin (pi/2 - x)

    But what else could I do to simply further?
    Use the sin addition formulae

    (Original post by coolstorybrother)
    P.s Could someone please clarify some of the terminology to me please? I believe an inverse function is when you reflect a function in y=x? so the inverse of y=sin x is y=sin^-1 x ? However Sin^-1 x =/= 1/sin x ? so Cosec is not an inverse function of sin x?
    Yeah that's right. The "-1" terminology is confusing. sin^-1 is the inverse function, but cosec is the multiplicative inverse of sin. Just like division is the inverse of multiplication, but the reciprocal 1/x is the multiplicative inverse of x.

    A better example is  f(x) = x^3 . The inverse function is  f^{-1}(x) = x^{1/3} , but the multiplicative inverse is  (f(x))^{-1} = \frac{1}{x^3}
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    (Original post by dantheman1261)
    It's the definition - literally, cosec is just a quick way of writing 1/sin

    aah okay

    Use the sin addition formulae

    hhmm I thought it wasn't something straight forward, I have not covered that yet and it wasn't in my book, could you please explain it?


    Yeah that's right. The "-1" terminology is confusing. sin^-1 is the inverse function, but cosec is the multiplicative inverse of sin. Just like division is the inverse of multiplication, but the reciprocal 1/x is the multiplicative inverse of x.

    A better example is  f(x) = x^3 . The inverse function is  f^{-1}(x) = x^{1/3} , but the multiplicative inverse is  (f(x))^{-1} = \frac{1}{x^3}

    I think i get it:
    sin^-1 x = inverse
    cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
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    (Original post by coolstorybrother)

    hhmm I thought it wasn't something straight forward, I have not covered that yet and it wasn't in my book, could you please explain it?
    Ahh - actually, sin(pi/2 - x) can be immediately rewritten as a different trigonometric function (I can't be any more clear without totally giving it away )

    (Original post by coolstorybrother)

    I think i get it:
    sin^-1 x = inverse
    cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
    That's it
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    (Original post by dantheman1261)
    Ahh - actually, sin(pi/2 - x) can be immediately rewritten as a different trigonometric function (I can't be any more clear without totally giving it away )



    That's it
    aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
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    (Original post by coolstorybrother)
    aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
    You do not really need to consider the transformation

    sin(90-x) = cos(x)

    cos(90-x) = sin(x)

    Just from the triangles

    (used degrees to avoid needing latex)
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    (Original post by coolstorybrother)
    aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
    That's right
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    sec x is the inverse of cos x. BECAUSE, LOOK AT THE LETTER C, WHICH INDICATES COS X
    COSEC X IS THE INVERSE OF SIN X. BECAUSE, 1/sinx=cosecx or letter S INDICATES SIN X!
    So sin(pi/2-x)=cos(2-x)
    So cosec(pi/2 - x)=sec(2 - x)
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    (Original post by ECONMATHSMATHSMATH)
    sec x is the inverse of cos x. BECAUSE, LOOK AT THE LETTER C, WHICH INDICATES COS X
    COSEC X IS THE INVERSE OF SIN X. BECAUSE, 1/sinx=cosecx or letter S INDICATES SIN X!
    So sin(pi/2-x)=cos(2-x)
    So cosec(pi/2 - x)=sec(2 - x)
    I think You should to use the multiplicative inverse or more the reciprocal
    term for above.
    For functions the inverse, maybe inverse relation or inverse function, and this is another business.
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    (Original post by coolstorybrother)

    I think i get it:
    sin^-1 x = inverse
    cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
    I know loads of books use \sin^{-1} as the inverse function but yes it is ambiguous. In the end, why would \sin^{-2}=\csc^2 but \sin^{-1}\neq\csc?

    Instead, use \arcsin,\;\arccos,\;\arctan... as the inverse functions!

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