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I am trying to understand how to create an identity for $\sin \frac{x}{4}$.

There are a number of steps that I have not understood.

1) $\sin\left( \frac{x}{4}\right) = \sin\left( \frac{\frac{x}{2}}{2}\right)$

Question: How are these two equivalent?

Thank you for the answer given to this part.

2) I know it has something to do with $\pm \sqrt \frac{1 - \cos \frac{x}{2}}{2}$

Question: How is this derived?

3) This is then turned into $\pm \sqrt{\frac{1 \mp \sqrt{ \frac{1 + \cos x}{2}}}{2}}$

Question: Where does the first $1$ come from in the problem(right before the $\mp$)?

Question: In $\frac{ 1 + \cos x}{2}$ how does the $2$ end up in the denominator?

4) Finally the answer is given as $\pm \sqrt{ \frac{\sqrt{2}\mp\sqrt{1 + \cos x}} {2\sqrt{2} } }$

Question: Where does the $\sqrt{2}$ come from in the beginning of the problem? How does it end up $2\sqrt{2}$ in the denominator?

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    _Regarding first question:_ Before studying trigonometry, it's suggested that you learn dividing fractions. :) – 2017-01-03
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    I'll do a review, but, a little help for now would be good. – 2017-01-03

1 Answers 1

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A quarter of a quantity is the same as half of half of the quantity.

$\frac x4 \equiv \frac {\left (\frac x2 \right)}{2} $

So $\sin \frac{x}{4} \equiv \sin \left( \frac {\left (\frac x2 \right)}{2} \right)$

For the rest, do you know about "double angle formulae"?

  • 0
    Oh, ok, so all that meant is $\frac{1}{2} / 2$ – 2017-01-03
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    Yes, that is what a fraction means. – 2017-01-03
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    Could, you answer the other questions? – 2017-01-03
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    I know a little about double angle formula, but I would have to look at the book. – 2017-01-03
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    A good answer that says enough with enough left over for the questioner to discover. – 2017-01-03
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    Nope, need futher explaining. I cannot find the answer. – 2017-01-03