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Assume that 0 < x < pi/2 and sin(x) = z

a.) Find a formula that gives the value of sin(x/2) in terms of z

b.) Corroborate the validity of the formula for these values of x:

  • pi/4
  • pi/3
  • pi/6

I know that it uses the first quadrant?

and sin(x/2) = +- sqrt((1-cos(x))/2)?

I don't get how I'm supposed to come up with a formula that give the value of sin(x/2) in terms of z though. Any suggestions? Thanks.

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It's going to look ugly... You have a formula relating sine and cosine, correct? –  Mike Apr 22 '13 at 18:55
    
What do you mean? I posted the info given. Do you mean the sin(x/2) part? –  ModdedLife Apr 22 '13 at 19:19

1 Answer 1

We derive the result from the more familiar double-angle identity $$\cos 2t=1-2\sin^2 t.$$ (Probably $\cos 2t=2\cos^2 t -1$ is even more familiar. We can then get to $1-2\sin^2 t$ by replacing $\cos^2 t$ with $1-\sin^2 t$.)

Let $t=x/2$. Then $$2\sin^2 (x/2)=1-\cos x.$$ Now $\cos x=\pm\sqrt{1-\sin^2 x}$. But if our angle is between $0$ and $\pi/2$, then $\cos x$ is positive, and so is $\sin(x/2)$.

So we get $2\sin^2(x/2)=1-\sqrt{1-z^2}$, and therefore $$\sin(x/2)=\sqrt{\frac{1-\sqrt{1-z^2}}{2}}.\tag{$1$}$$

Example: Let $x=\frac{\pi}{4}$. Then $z=\sin(\pi/4)=\frac{1}{\sqrt{2}}$. Substitute in $(1)$. Note that $1-z^2=\frac{1}{2}$. So we get $$\sin(\pi/8)=\sqrt{\frac{1-\sqrt{1/2}}{2}}.$$ One could make a further algebraic simplification.

Remark: It is actually worthwhile to do algebraic simplifications. By repeating the calculation, we can successively get expressions for $\sin(\pi/16)$, $\sin(\pi/32)$, and so on forever. If we do, we get a beautiful expression for $\pi$. In a quite different form, the idea goes back to Archimedes.

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Hmm. Why did you start with cosine? –  ModdedLife Apr 22 '13 at 19:24
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Confession time (you can check the edits!). I started with $\sin 2t=2\sin t\cos t$, replaced $\cos t$ by $\sqrt{1-\sin^2 t}$, squared both sides. We get a quadratic equation in $\sin^2 t$, which I solved. Straightforward, but kind of ugly. I wanted a nicer argument, so rewrote. –  André Nicolas Apr 22 '13 at 19:30
    
Hrmph. Not sure I'm following lol –  ModdedLife Apr 22 '13 at 19:54
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Well, part of it is still visible if you go to Edit 3 (click on the "edited x minutes ago" link. Or trust me, it was uglier, but more natural. –  André Nicolas Apr 22 '13 at 19:57
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I assume you are now asking about the answer as it is now. I will add a specific computation for one of the questions you were asked. Less than $10$ minutes! –  André Nicolas Apr 22 '13 at 20:03

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