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We find the exact value
of sine of (5 pi twelfths).
It
equals sine of
(3 pi twelfths plus
2 pi twelfths). Because
three pi plus 2 pi is the same as 5 pi.
If we reduce,
you will have sine of
(pi fourths plus pi sixths). And these 2 are
both common angles.
We use the sum identity for sine which says
sine of alpha plus
beta is the same as
sine, alpha, cosine, beta,
plus, cosine, alpha, sine,
beta.
We continue. It equals
sine of the first angle, pi
fourths, cosine of
the second angle, pi sixths,
plus cosine of the first angle,
pi fourths times sine of
pi sixths.
These are common angles.
It equals root 2, over 2,
times, root 3, over 2,
plus root 2, over 2, again,
times one half.
Pull out the common factors, we get
root 2 times the sum, (root 3, plus one)
over 4. This is our
exact value.