How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8})$ $sin(pi/12) * cos(( 5 pi)/8 )$ without products of trigonometric functions?

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8})$ $sin(pi/12) * cos(( 5 pi)/8 )$ without products of trigonometric functions?

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Answer 1 · 2016-02-27T14:35:17

$sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8}) = - \frac{\sqrt{(1 - \frac{\sqrt{3}}{2}) (1 - \frac{\sqrt{2}}{2})}}{2}$ $sin(pi/12)*cos((5pi)/8)=-(sqrt((1-sqrt3/2)(1-sqrt2/2)))/2$

Explanation:

$[1] sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8})$ $[1]" "sin(pi/12)*cos((5pi)/8)$

Think of $\frac{π}{12}$ $pi/12$ as $\frac{\frac{π}{6}}{2}$ $(pi/6)/2$ .

$[2] = sin (\frac{\frac{π}{6}}{2}) \cdot cos (\frac{5 π}{8})$ $[2]" "=sin((pi/6)/2)*cos((5pi)/8)$

Half angle identity: $sin (\frac{x}{2}) = \pm \sqrt{\frac{1 - cos (x)}{2}}$ $sin(x/2)=+-sqrt((1-cos(x))/2)$
** $\frac{π}{12}$ $pi/12$ is in the first quadrant, and sine is positive in the first quadrant. Therefore, we know that $sin (\frac{\frac{π}{6}}{2}) = + \sqrt{\frac{1 - cos (\frac{π}{6})}{2}}$ $sin((pi/6)/2)=+sqrt((1-cos(pi/6))/2)$

$[3] = \sqrt{\frac{1 - cos (\frac{π}{6})}{2}} \cdot cos (\frac{5 π}{8})$ $[3]" "=sqrt((1-cos(pi/6))/2)*cos((5pi)/8)$

Evaluate $cos (\frac{π}{6})$ $cos(pi/6)$

$[4] = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot cos (\frac{5 π}{8})$ $[4]" "=sqrt((1-sqrt3/2)/2)*cos((5pi)/8)$

Think of $\frac{5 π}{8}$ $(5pi)/8$ as $\frac{\frac{5 π}{4}}{2}$ $((5pi)/4)/2$ .

$[5] = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot cos (\frac{\frac{5 π}{4}}{2})$ $[5]" "=sqrt((1-sqrt3/2)/2)*cos(((5pi)/4)/2)$

Half angle identity: $cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos (x)}{2}}$ $cos(x/2)=+-sqrt((1+cos(x))/2)$
** $\frac{5 π}{8}$ $(5pi)/8$ is in the second quadrant, and cosine is negative in the second quadrant. Therefore, we know that $cos (\frac{\frac{5 π}{4}}{2}) = - \sqrt{\frac{1 + cos (\frac{5 π}{4})}{2}}$ $cos(((5pi)/4)/2)=-sqrt((1+cos((5pi)/4))/2)$

$[5] = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot ⎛ ⎜ ⎜ ⎜ ⎝ - \sqrt{\frac{1 + cos (\frac{5 π}{4})}{2}} ⎞ ⎟ ⎟ ⎟ ⎠$ $[5]" "=sqrt((1-sqrt3/2)/2)*(-sqrt((1+cos((5pi)/4))/2))$

Evaluate $cos (\frac{5 π}{4})$ $cos((5pi)/4)$

$[6] = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot ⎛ ⎜ ⎜ ⎜ ⎝ - \sqrt{\frac{1 + (- \frac{\sqrt{2}}{2})}{2}} ⎞ ⎟ ⎟ ⎟ ⎠$ $[6]" "=sqrt((1-sqrt3/2)/2)*(-sqrt((1+(-sqrt2/2))/2))$

Simplify.

$[7] = - \frac{\sqrt{(1 - \frac{\sqrt{3}}{2}) (1 - \frac{\sqrt{2}}{2})}}{2}$ $[7]" "=color(blue)(-(sqrt((1-sqrt3/2)(1-sqrt2/2)))/2)$

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8})$ $sin(pi/12) * cos(( 5 pi)/8 )$ without products of trigonometric functions?

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Explanation:

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How do you express sin(pi/12) * cos(( 5 pi)/8 ) sin(π12)⋅cos(5π8)sin(pi/12) * cos(( 5 pi)/8 ) without products of trigonometric functions?

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How do you express $sin (\frac{π}{12}) \cdot cos (\frac{5 π}{8})$ $sin(pi/12) * cos(( 5 pi)/8 )$ without products of trigonometric functions?