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.
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??
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