Compound Angle Formula: Solving Double Angle Equations

[Farnaz]

TL;DR Summary

sin(19pi/2)=?

Hi, I am confused about how to handle the double angle formula. For example, sin(19pi/12)= sin(9pi/12+10pi/12) but there can be many other options too. like sin(18pi/12+pi/12) or sin(15pi/12+4pi/12)..every time I am getting different answers. Can anyone please how find the right one? Thanks

 

[Doc Al]

Can you give an example where you get a different answer? And what formula are you using to expand sin(A +B)?

 

[Mark44]

Summary:: sin(19pi/2)=?

You wrote $\sin(\frac{19\pi}2)$ above, but are asking about $\sin(\frac{19\pi}{12})$ below.

Hi, I am confused about how to handle the double angle formula. For example, sin(19pi/12)= sin(9pi/12+10pi/12) but there can be many other options too. like sin(18pi/12+pi/12) or sin(15pi/12+4pi/12)..every time I am getting different answers. Can anyone please how find the right one? Thanks

The "right one" is a combination that you can do something with, assuming the answers are to be exact rather than approximate ones.
For example, $\sin(9\pi/12 + 10\pi/12) = \sin(3\pi/4 + 5\pi/6)$ is better than your other choices here because the angles are such that the sine and cosine terms all have exact answers.

 

[lurflurf]

^I don't know what you mean by exact, none of the above is approximation.

I don't understand the op either, but I would guess he/she has two equal numbers and did not recognize them as such.

For example something like
$$\sin \left( \frac{\pi}{12} \right)=\frac{\sqrt{2-\sqrt{3}}}{2}$$
$$\sin \left( \frac{\pi}{12} \right)=\frac{\sqrt{6}-\sqrt{2}}{4}$$
these are equal even though they look different

 

[Mark44]

I don't know what you mean by exact, none of the above is approximation.

I assume you're quoting what I said. The context for my remark was that the sines and cosines of certain angles can be calculated exactly and simply; for example, $\sin(3\pi/4) = \sqrt 2/2$ and ##\cos(2\pi/3) = -1/2. The trig functions of many other angles don't lend themselves such straightforward computation.

 

[mfb]

I would start with sin(19pi/12) = sin(pi + 7pi/12) = ... which leads to nice simple expressions with a suitable way to write 7pi/12 as sum.