How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{12})$ $sin(pi/12) * cos((5 pi)/12 )$ without products of trigonometric functions?

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{12})$ $sin(pi/12) * cos((5 pi)/12 )$ without products of trigonometric functions?

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Answer 1 · 2016-02-26T04:19:17

$\frac{2 - \sqrt{3}}{4}$ $(2 - sqrt3)/4$

Explanation:

On the unit circle, using property of complementary arcs:
$cos (\frac{5 π}{12}) = sin (\frac{π}{2} + \frac{5 π}{12}) = sin (\frac{11 π}{12}) = \frac{sin π}{12}$ $cos ((5pi)/12) = sin (pi/2 + (5pi)/12) = sin ((11pi)/12) = sin pi/12$
Product $P = sin (\frac{π}{12}) . cos (\frac{5 π}{12}) = {sin}^{2} (\frac{π}{12})$ $P = sin (pi/12).cos ((5pi)/12) = sin^2 (pi/12)$ .
Find ${sin}^{2} (\frac{π}{12}) .$ $sin^2 (pi/12).$
Use trig identity: $cos (\frac{π}{6}) = 1 - 2 {sin}^{2} (\frac{π}{12}) = \frac{\sqrt{3}}{2}$ $cos (pi/6) = 1 - 2sin^2 (pi/12) = sqrt3/2$
$2 {sin}^{2} (\frac{π}{12}) = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}$ $2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2$
$P = {sin}^{2} (\frac{π}{12}) = \frac{2 - \sqrt{3}}{4}$ $P = sin^2 (pi/12) = (2 - sqrt3)/4$

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{12})$ $sin(pi/12) * cos((5 pi)/12 )$ without products of trigonometric functions?

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Explanation:

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How do you express sin(π12)⋅cos(5π12)sin(pi/12) * cos((5 pi)/12 ) without products of trigonometric functions?

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{12})$ $sin(pi/12) * cos((5 pi)/12 )$ without products of trigonometric functions?