Wikipedia:Reference desk/Archives/Mathematics/2008 February 6

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February 6

Trig Proof

I'm stuck on the following proof. Could anyone help me out? ${\frac {cosx+1+sinx}{cosx+1-sinx}}={\frac {1+sinx}{cosx}}$ --Sturgeonman (talk) 00:58, 6 February 2008 (UTC)

Got it. Never mind. -- Sturgeonman (talk) 01:34, 6 February 2008 (UTC)

BTW, you should put a backslash before common mathematical functions like \cos and \sin in LaTeX. It makes it look much better:

${\frac {\cos x+1+\sin x}{\cos x+1-\sin x}}={\frac {1+\sin x}{\cos x}}$ —Keenan Pepper 06:04, 6 February 2008 (UTC)

Two unrelated calculus questions

1) I'm being asked to evaluate the indefinite intergal ∫sin² 2x and for some reason, I don't know how to proceed ... all my calculations have been fruitless and led to culs-de-sac. Could anyone at least give me a hint in the right direction? (I don't think the [First] Fundamental Theorem of Calculus comes into play, since that's for definite integrals, if I'm not mistaken.)

2) For computing a Riemann sum using the midpoint method, my teacher recommends staying away from the formula and using a simple, straightforward nonformulaic approach. It's something like taking the midpoint of each interval and adding f(midpoint₁) + f(midpoint₂) + ... + f(midpoint_n-1) + f(midpoint_n). Will this give me the right answer, or do I have to multiply through by ∆x/2 or something like that, like when finding a trapezoidal sum?

Thanks! —anon —Preceding unsigned comment added by 141.155.57.220 (talk) 05:12, 6 February 2008 (UTC)

There is a version of the FTOC that applies to indefinite integrals, but it's true that in this case you need something a little more - and that something a little more is a trick called a double-angle formula. Put simply, do you know an identity that lets you write cos 2x in terms of sin x? And, having done so, can you rearrange that into something you can use in your first integral?

As for your second question, try thinking about it conceptually - a Riemann sum is essentially an approximation of the area of a bunch of rectangles. The area of a rectangle is the product of the two side lengths, so there's a reasonable chance you should be multiplying something by something to get your results. Confusing Manifestation(Say hi!) 05:25, 6 February 2008 (UTC)

The problem is not the 2x term, which can be dealt with by u-substitution. It's the squaring of the sine. There are power reduction formulae at the trig identities article. Black Carrot (talk) 09:08, 6 February 2008 (UTC)

I think you misunderstood ConMan's post. Note that he specified "cos 2x", which can be expanded using a $\sin ^{2}x$. This formula is memorized more commonly than the explicit formula for $\sin ^{2}x$ itself, and is thus a good starting point for the calculation. -- Meni Rosenfeld (talk) 23:09, 8 February 2008 (UTC)

(1) $\sin ^{2}(2x)={\tfrac {\sin ^{2}(2x)+\sin ^{2}(2x)}{2}}={\tfrac {1-\cos ^{2}(2x)+\sin ^{2}(2x)}{2}}={\tfrac {1-\cos(4x)}{2}}={\tfrac {1}{2}}-{\tfrac {1}{2}}\cos(4x)$
HTH. CiaPan (talk) 10:02, 6 February 2008 (UTC)

(2) Of course you should multiply each f(midpoint_i) by Δx, which is the n-th part of the integration interval. Putting all Δx-es outside the parens makes the integral approximation = IntervalLength/n × sum of f(midpoint_i). --CiaPan (talk) 09:57, 6 February 2008 (UTC)