# Thread: What is the exact value of sin 12 degrees?

1. ## What is the exact value of sin 12 degrees?

What is the exact value of sin 12 degrees?

Only hint I was given was to use compound angle formula. Had this on the test but I left it blank, now I just want to know the solution.

2. Originally Posted by ISeriouslyNeedHelp
What is the exact value of sin 12 degrees?

Only hint I was given was to use compound angle formula. Had this on the test but I left it blank, now I just want to know the solution.
12 degrees is $\displaystyle \frac{\pi}{15}$

so one way to do this is to use

$\displaystyle sin(U-V) - sinUcosV-cosUsinV$

divide up $\displaystyle \frac{\pi}{15}$ into U and V

for example $\displaystyle \frac{3\pi}{5} - \frac{\pi}{3} = \frac{\pi}{15}$

then use the difference formula...

can you take it from there??

3. Originally Posted by bigwave
12 degrees is $\displaystyle \frac{\pi}{15}$

so one way to do this is to use

$\displaystyle sin(U-V) - sinUcosV-cosUsinV$

divide up $\displaystyle \frac{\pi}{15}$ into U and V

for example $\displaystyle \frac{3\pi}{5} - \frac{\pi}{3} = \frac{\pi}{15}$

then use the difference formula...

can you take it from there??
Now of course there's the new problem of finding $\displaystyle \sin \frac{3 \pi}{5}$ and $\displaystyle \cos \frac{3 \pi}{5}$ ....

I suspect there's more to the original question than was posted.

4. There is nothing more to this question. It is simply to find the exact value of sin 12 degrees. Only thing I could think of was sin(60/5), but these odd numbers really mess with my head.

5. Originally Posted by ISeriouslyNeedHelp
What is the exact value of sin 12 degrees?

Only hint I was given was to use compound angle formula. Had this on the test but I left it blank, now I just want to know the solution.
To do this you need:

$\displaystyle \sin(12^{\circ})=\sin(30^{\circ}-18^{\circ})$

We know the values of the trig functions for $\displaystyle 30^{\circ}$ and so if we also know the trig function values for $\displaystyle 18^{\circ}$ we are home and dry.

We can get the values we need for the $\displaystyle \sin$ and $\displaystyle \cos$ of $\displaystyle 18^{\circ }$ from the geometry of a regular pentagon (which we could have guessed as we are dealing with integer fractions of $\displaystyle 72^{\circ}=360^{\circ}/5)$, where we find:

$\displaystyle \sin(18^{\circ})=\frac{1}{2\Phi}$

where $\displaystyle \Phi=\frac{1+\sqrt{5}}{2}$ is the Golden number.

CB

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