What is the exact value of $sin (\frac{7 π}{12}) - sin (\frac{π}{12})$ $sin((7pi)/12)-sin(pi/12)$ ?

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Answer 1 · 2015-02-13T22:56:44

$sin( (7Pi)/12) − sin(Pi/12) = 1/sqrt(2)$

One of the standard trig. formulas states:
$sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )$

So
$sin( (7Pi)/12) − sin(Pi/12)$
$= 2 sin( ((7Pi)/12 - (pi)/12)/2 ) cos( ((7Pi)/12 + (Pi)/12)/2 )$
$= 2 sin( Pi/4 ) cos( Pi/3 )$

Since $sin(Pi/4) = 1/( sqrt(2) )$

and $cos ( (2Pi)/3) = 1/2$

$2 sin( Pi/4 ) cos( (2Pi)/3 )$
$= (2) (1/(sqrt(2))) (1/2)$
$= 1/sqrt(2)$

Therefore
$sin( (7Pi)/12) − sin(Pi/12) = 1/sqrt(2)$

What is the exact value of sin((7pi)/12)-sin(pi/12)sin(7π12)−sin(π12)sin((7pi)/12)-sin(pi/12)?