Hey there Sign in to join this conversationNew here? Join for free

sec, cosec and cot question

Announcements Posted on
Study Help needs new mods! 14-04-2014
Post on TSR and win a prize! Find out more... 10-04-2014
    • Thread Starter
    • 17 followers
    Offline

    ReputationRep:
    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?
    • 1 follower
    Offline

    ReputationRep:
    (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}
    • Thread Starter
    • 17 followers
    Offline

    ReputationRep:
    (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?
    • 1 follower
    Offline

    ReputationRep:
    (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
    • Thread Starter
    • 17 followers
    Offline

    ReputationRep:
    (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.
    • 40 followers
    Offline

    ReputationRep:
    (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)
    • 1 follower
    Offline

    ReputationRep:
    (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
    • 1 follower
    Offline

    ReputationRep:
    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)
    • 3 followers
    Offline

    ReputationRep:
    (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.
    • 18 followers
    Offline

    ReputationRep:
    (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!

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?

    this is what you'll be called on TSR

  2. this can't be left blank
    this email is already registered. Forgotten your password?

    never shared and never spammed

  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By completing the slider below you agree to The Student Room's terms & conditions and site rules

  2. Slide the button to the right to create your account

    Slide to join now Processing…

    You don't slide that way? No problem.

Updated: August 6, 2012
Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.