man0005's specialist question thread :)

[kamil9876]

Don't know if there is any quicker way yet but you can always do it by brute force by just expanding sin(3x)=sin(x+2x) and likewise sin(5x)=sin(3x+2x)... same for the cosines, then just cleaning up a big mess.

I think a slightly quicker way may be to write sin(x)=sin(3x-2x) and expand that out, write sin(5x)=sin(3x+2x) and expand that out.

Here is what you get for the numerator:







Now do a similair thing for the denominator and hopefully it will work out (that is expand out and )

[evaever]

got a really good one for you kamil d:

(sin x + sin 3x + sin 5x)
____________________      =   tan 3x

(cos x + cos 3x + cos 5x)

I'm not Kamil. I'll do it anyway.

cisx+cis3x+cis5x = (cosx+cos3x+cos5x)+i(sinx+sin3x+sin5x)
kcis3x+cis3x=(cosx+cos3x+cos5x)+i(sinx+sin3x+sin5x)
(k+1)cis3x=(cosx+cos3x+cos5x)+i(sinx+sin3x+sin5x)
arg of left = arg of right
3x=tan^-1 (sinx+sin3x+sin5x)/(cosx+cos3x+cos5x)
tan3x=(sinx+sin3x+sin5x)/(cosx+cos3x+cos5x)

[man0005]

O_O
evaever are we meant to know how to do it like that in spesh D:
cause i have no idea what you did LOL

[kamil9876]

evaever did not define in the solution so I don't know what he/she did entirely.

Here is another solution using complex numbers(using the idea in my previous solution to change 5x to 3x+2x and x to 3x-2x):

let



But you see that cis(-2x)+cis(2x) is a real number. Hence we have that cis(x)+cis(3x)+cis(5x) is a real number multiply of cis(3x), hence which is the result.

edit: probably won't be expected to do it like this, brute force works and that's how most of specialist rolls.

generalisation: suppose that is an arithmetic sequence with arithmetical average , then (if it is defined)

[man0005]

oh okay, lol someone eventually googled it and there some random formula to do it. wasted all my time in the end

anyway i have a quick question
can't seem to get this one
Let sin x = 0.75, x is an element of pie, 3pie/2, Find correct to 2 decimal places sin 1/2 x
(We're meant to use identities to get it, not calculator)
sorry if this seems like a silly question, i'm a little tired today

[kamil9876]

oh okay, lol someone eventually googled it and there some random formula to do it. wasted all my time in the end

what do you mean? it is doable with just the identities you know as shown in the first post after the solution. The complex number solution was just a more slick way that you don't really need to know and we didn't use the generalisation to solve it, it is just something I noticed after looking back at my complex number solution.

Anyway for your question:

I assume that is ? If so let you see that:



Still this would require a little bit of calculator work but only for the square roots that you need to find cos(2y) using pythagoras as well as square rooting at the end (so you can find an answer with surds basically.

[man0005]

Can someone help me with this please

The amount of pollution (p), t centuries after 1850 (ie t = 0 is 1850) is given by the rule:
P = aTan^-1(bt)

Find the percentage increase in pollution from 1900 to 2000.
in 1900, P = 2.657, in 1950, P = 4.5

[Mao]

Substituting (t,P)=(0.5,2.657) and (1.0,4.5), we arrive at

2.657=aTan^-1[0.5b]
4.5=aTan^-1(b)

eqn1/eqn2:
2.657Tan^-1(b)=4.5Tan^-1[0.5b]

Plot 2.657Tan^-1(b) and 4.5Tan^-1[0.5b], find that the intersection is at b=1.00082. Further substitution gives a=5.72658

If the question provided exact figures, the answer would be b=1 and a=18/pi

[man0005]

How would you do this D:
Split into partial fractions

(-x-10)
__________

(x^2 - 4x -12)

[Mao]

Equating coefficients

[man0005]

ohh right, sorry
what about splitting:

4x^2 - 21x + 23
___________________
x^2 - 6x + 9

[pi]

Just a hint: Try using long division first

[m@tty]

Don't use long division on that..

Just recognise that there are 4 x^2 on the numerator, thus try and get 4 of the denominator separate to divide.

[pi]

^^Thats the way I would have done it too, I was just giving a stock-standard approach. Of course, visualising a factor is always better and neater man0005, just that its not in the texts for some reason...

[man0005]

thanks for that guys
i really didnt want to use long division