How do you express $cos (\frac{π}{3}) \cdot sin (\frac{15 π}{8})$ $cos(pi/ 3 ) * sin( ( 15 pi) / 8 )$ without using products of trigonometric functions?

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{15 π}{8})$ $cos(pi/ 3 ) * sin( ( 15 pi) / 8 )$ without using products of trigonometric functions?

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Answer 1 · 2016-02-25T04:18:26

$P = \frac{1}{2} - (\frac{\sqrt{2 + \sqrt{2}}}{2})$ $P = 1/2 - (sqrt(2 + sqrt2)/2)$

Explanation:

$P = cos (\frac{π}{3}) . sin (\frac{15 π}{8})$ $P = cos (pi/3).sin ((15pi)/8)$
$cos (\frac{π}{3}) = \frac{1}{2}$ $cos (pi/3) = 1/2$
$sin (\frac{15 π}{8}) = sin (2 π - \frac{π}{8}) = - \frac{sin π}{8} .$ $sin ((15pi)/8) = sin (2pi - pi/8) = - sin pi/8.$
Find $sin (\frac{π}{8})$ $sin (pi/8)$ by using trig identity:
$cos (\frac{π}{4}) = 2 {cos}^{2} (\frac{π}{8}) - 1 = \frac{\sqrt{2}}{2}$ $cos (pi/4) = 2cos^2 (pi/8) - 1 = sqrt2/2$
$2 {cos}^{2} (\frac{π}{8}) = 1 + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2}$ $2cos^2 (pi/8) = 1 + sqrt2/2 = (2 + sqrt2)/2$
${cos}^{2} {(\frac{π}{8})}^{2} = \frac{2 + \sqrt{2}}{4}$ $cos^2 (pi/8)^2 = (2 + sqrt2)/4$
$cos (\frac{π}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$ $cos (pi/8) = (sqrt(2 + sqrt2))/2$ (cos (pi/8) is positive.)
$P = \frac{1}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2}$ $P = 1/2 - (sqrt(2 + sqrt2))/2$

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{15 π}{8})$ $cos(pi/ 3 ) * sin( ( 15 pi) / 8 )$ without using products of trigonometric functions?

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{15 π}{8})$ $cos(pi/ 3 ) * sin( ( 15 pi) / 8 )$ without using products of trigonometric functions?