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    (Original post by A Slice of Pi)
    Heres a quick question you might want to try for extra practice...

    Use the method of differences to show that

    \mathrm{cosec}\hspace{3pt}2 + \mathrm{cosec}\hspace{3pt}4 + \mathrm{cosec}\hspace{3pt}8 + \ldots + \mathrm{cosec}\hspace{3pt}2^n = \cot 1 - \cot 2^n

    (Hint: Think about trig identities linking cosec and cot)
    Ah, your hint gave it away.

    It's not hard to prove that \tan \frac{x}{2} = \frac{\sin x}{1+\cos x} from which you argue that \cot \frac{x}{2} = \csc x - \cot x and hence \csc x = \cot \frac{x}{2} - \cot x.

    So your sum is: \cot 1 - \cot 2 + \cot 2 - \cot 4 + \cdots  + \cot 2^{n-1} - \cot 2^n which telescopes down to \cot 1 - \cot 2^n.
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    I feel bad for stealing the above question, so I'll contribute one:

    \displaystyle

\begin{equation*} \sum_{n=1}^m 2^n \tan (2^n \theta) \end{equation*}
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    https://8dedc505ac3fba908c50836f5905...%20Edexcel.pdf

    Q5b - i get the 1/2 but get completely lost trying to find what n terms wouldnt cancel?
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    Anyone wanna help me answer these questions. I cant do anything until I get these cleared out and time is running out. So reposted. Zacken can you help you seem tobe able to solve anything
    Inequalities:
    -Do you need to know how to draw the 1/x , x / xsquared graphs etc. If so, can anyone explain to me how to draw the graphs once you get the x values and then how do you work out the inequalities for those type of graphs.
    -How do you do questions 1 and 2 on page 9 (I know how to get the x values for 1 but cant do the rest and I cant do 2 at all)

    Complex numbers:
    - Page 49 question 6. How to draw the locus without solving the equation for those questions when theres a number outside the |z|.
    - Page 59 question 1b??

    Polar coordinates:
    -On pages 133 and 134 when sketching the curves, the choose different values for theta. How do you know which values to choose.
    - Page 143 question 4. How do you solve it when its r squared.

    Thank you
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    For questions like this one does it matter whether you give your answers with theta between -pi and pi or between 0 and 2pi?
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    (Original post by lkara)
    For questions like this one does it matter whether you give your answers with theta between -pi and pi or between 0 and 2pi?
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    No.
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    (Original post by fpmaniac)
    Anyone wanna help me answer these questions. I cant do anything until I get these cleared out and time is running out. So reposted. Zacken can you help you seem tobe able to solve anything
    Inequalities:
    -Do you need to know how to draw the 1/x , x / xsquared graphs etc. If so, can anyone explain to me how to draw the graphs once you get the x values and then how do you work out the inequalities for those type of graphs.
    -How do you do questions 1 and 2 on page 9 (I know how to get the x values for 1 but cant do the rest and I cant do 2 at all)
    You should be able to sketch rational functions. It's not hard, all you need to note is that \frac{1}{x} is asymptotic to the y-axis for small values of x and then curves downwards steeply before levelling out and moving down towards the x-axis asymptotically for large values of x. That's the behaviour in the first quadrant; behaviour in the third quadrant is basically the same although mirrored.

    \frac{1}{x^2} is pretty much the same except it's in the first and second quadrant, i.e: symmetric in the y-axis.

    I don't own a textbook so can't help without pictures or something.
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    (Original post by imnoteinstein)
    https://8dedc505ac3fba908c50836f5905...%20Edexcel.pdf

    Q5b - i get the 1/2 but get completely lost trying to find what n terms wouldnt cancel?
    Not easy to explain other than looking at what they've done.

    the 'middle term' for r = k is cancelled out by combining the 'last term' in r = k-1 and the 'first term' in r = k +1, so you work through up to r = 4 and you see that the only term at the start that isn't cancelling out is 1/2.

    Then you work towards the end (starting from n-2).

    So for r=n-2 everything cancels out (as you have the 'first term' below it and the 'last term' above it so the middle term is cancelled out, the 'first term' in r=n-2 is used up in cancelling out the previous 'middle term' and the last one is used in cancelling out the 'middle term' in r = n-1.



    Similar logic in r=n-1 for the first and second term, but the third term is only half of the middle term in r = n so it only 'half cancels' it.

    which is why you get 1/(n+1) - 2/(n+1) = -1/(n+1) as one of the terms in the summation.



    and then for r=n the 'middle term' half cancels (as before) and the first one is used up, and the last one does not do anything as there needs to be something below it.
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    (Original post by Zacken)
    You should be able to sketch rational functions. It's not hard, all you need to note is that \frac{1}{x} is asymptotic to the y-axis for small values of x and then curves downwards steeply before levelling out and moving down towards the x-axis asymptotically for large values of x. That's the behaviour in the first quadrant; behaviour in the third quadrant is basically the same although mirrored.

    \frac{1}{x^2} is pretty much the same except it's in the first and second quadrant, i.e: symmetric in the y-axis.

    I don't own a textbook so can't help without pictures or something.
    1. Solve the inequality: |x² - 7| < 3(x+1) ....... if the 3 is outside the bracket do you treat it like enlargement
    2.
    (x² /|x| + 6) < 1 How do you solve it


    3. How have they calculated the asymptotes for
    y = 7x/3x+1


    Complex numbers
    1. how do you sketch the locus without solving it first: |z+3| = 3|z-5|
    2. For the transformation w=z+4+3i sketch on an argand diagram the locus of w when z lies on the half line arg z = pi/2 ,
    3. " " " " w = z-1+2i when z lies on the line y=2x

    Polar:
    1. Find the polar coordinates of the points on r²=a²sin2θ where the tangent is perpendicular to the initial line. ( dont know how to solve with r² so help me how to start pls.)
    2. Sketching curves:
    For r=a(1+cosθ) Thhey have chosen θ to be 0 , pi/2, pi, 3pi/2, and 2pi
    But for r=sin3θ they considered 0≤θ≤pi/3 , 2pi/3≤θ≤pi and 4pi/3≤θ≤5pi/3 and they chose the values of θ to be 0, pi/6 and pi/3. How do you know which values to choose an which inequalities to consider.

    Thanks
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    (Original post by Zacken)
    Ah, your hint gave it away.

    It's not hard to prove that \tan \frac{x}{2} = \frac{\sin x}{1+\cos x} from which you argue that \cot \frac{x}{2} = \csc x - \cot x and hence \csc x = \cot \frac{x}{2} - \cot x.

    So your sum is: \cot 1 - \cot 2 + \cot 2 - \cot 4 + \cdots  + \cot 2^{n-1} - \cot 2^n which telescopes down to \cot 1 - \cot 2^n.
    How would you go about proving
    \tan \frac{x}{2} = \frac{\sin x}{1+\cos x}

    ?
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    (Original post by Nikhilm)
    How would you go about proving
    \tan \frac{x}{2} = \frac{\sin x}{1+\cos x}

    ?
    The obvious way: \tan \frac{x}{2} = \frac{\frac{1}{2}\sin x}{\cos \frac{x}{2}\cos \frac{x}{2}} = \frac{\sin x}{2\cos^2 \frac{x}{2}} = \frac{\sin x}{1 + \cos x}
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    Can anyone help me with a few questions:

    1a. Find the value of z which satisfies both |z+2| = |2z-1| and arg z = pi/4
    b. Hence shade in the region R on an Argand diagram which satisfies both |z+2| ≥ |2z-1| and pi/4≤arg≤ pi

    2. Given that |z+1-i| = 1 find the greatest and least values of |z-1|
    3. Find the ccartesian equation of the locus of z when z-z* = 0 ...... what does the star mean
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    (Original post by Zacken)
    The obvious way: \tan \frac{x}{2} = \frac{\frac{1}{2}\sin x}{\cos \frac{x}{2}\cos \frac{x}{2}} = \frac{\sin x}{2\cos^2 \frac{x}{2}} = \frac{\sin x}{1 + \cos x}
    Ah yeah, I wouldn't have thought about using half angles for that question
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    (Original post by Nikhilm)
    Ah yeah, I wouldn't have thought about using half angles for that question
    Why not? The presence of you going from \csc 2 to \cot 1 on the LHS should have been screaming half angles. As were the powers of 2.
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    Does anyone have any good sources for the general way to deal with plane transformations?

    Z plane to W plane stuff?
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    (Original post by Nikhilm)
    Ah yeah, I wouldn't have thought about using half angles for that question
    I don't see how one can come up with that trig identity without working backwards lol
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    (Original post by Pyslocke)
    I don't see how one can come up with that trig identity without working backwards lol
    Zacken


    Ah yeah. They would probably ask you to prove something in a part (a) to give a hint on the approach
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    (Original post by Pyslocke)
    I don't see how one can come up with that trig identity without working backwards lol
    (Original post by Nikhilm)
    Zacken


    Ah yeah. They would probably ask you to prove something in a part (a) to give a hint on the approach
    At A-Level yeah. In general, it's pretty intuitive. Especially given the hint.
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    Anyone got tips on how to turn the equation into partial fractions the easier way? I still do it the C3/C4 way which takes much more time.
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    (Original post by Zacken)
    At A-Level yeah. In general, it's pretty intuitive. Especially given the hint.
    Yup hopefully! Btw where do you find all these obscure questions from?
 
 
 
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