How do you express $sin (\frac{π}{4}) \cdot sin (\frac{11 π}{12})$ $sin(pi/ 4 ) * sin( ( 11 pi) / 12 )$ without using products of trigonometric functions?

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How do you express $sin (\frac{π}{4}) \cdot sin (\frac{11 π}{12})$ $sin(pi/ 4 ) * sin( ( 11 pi) / 12 )$ without using products of trigonometric functions?

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Answer 1 · 2016-04-09T13:29:37

$(- \frac{\sqrt{2}}{2}) sin (\frac{π}{12})$ $(-sqrt2/2)sin (pi/12)$

Explanation:

$P = sin (\frac{π}{4}) . sin (\frac{π}{12})$ $P = sin (pi/4).sin (pi/12)$
Trig table --> $sin (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ $sin (pi/4) = sqrt2/2$
Trig table, trig unit circle, and property of supplement arcs -->
$sin (\frac{11 π}{12}) = sin (- \frac{π}{12} + \frac{12 π}{12}) = sin (- \frac{π}{12} + π) =$ $sin ((11pi)/12) = sin (-pi/12 + (12pi)/12) = sin (-pi/12 + pi) =$
$= - sin (\frac{π}{12}) .$ $= - sin (pi/12).$
The product can be expressed as:
$P = - (\frac{\sqrt{2}}{2}) sin (\frac{π}{12})$ $P = - (sqrt2/2)sin (pi/12)$
If required, you can find P's value by evaluating $sin (\frac{π}{12})$ $sin (pi/12)$ , using the trig identity: $cos (\frac{π}{6}) = 1 - 2 {sin}^{2} (\frac{π}{12}) = \frac{\sqrt{3}}{2}$ $cos (pi/6) = 1 - 2sin^2 (pi/12) = sqrt3/2$

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How do you express $sin (\frac{π}{4}) \cdot sin (\frac{11 π}{12})$ $sin(pi/ 4 ) * sin( ( 11 pi) / 12 )$ without using products of trigonometric functions?

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Explanation:

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How do you express $sin (\frac{π}{4}) \cdot sin (\frac{11 π}{12})$ $sin(pi/ 4 ) * sin( ( 11 pi) / 12 )$ without using products of trigonometric functions?