How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{4})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 4 )$ without using products of trigonometric functions?

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{4})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 4 )$ without using products of trigonometric functions?

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Answer 1 · 2016-05-21T04:12:34

$- (\frac{\sqrt{2}}{2}) sin (\frac{π}{8})$ $- (sqrt2/2)sin (pi/8)$

Explanation:

$P = sin (\frac{π}{8}) . cos (\frac{5 π}{4})$ $P = sin (pi/8).cos ((5pi)/4)$
Trig table and property of supplementary arcs -->
$cos (\frac{5 π}{4}) = cos (\frac{π}{4} + π) = - cos (\frac{π}{4}) = - \frac{\sqrt{2}}{2} .$ $cos ((5pi)/4) = cos (pi/4 + pi) = - cos (pi/4) = -sqrt2/2.$
Therefor, P can be expressed as
$P = - (\frac{\sqrt{2}}{2}) sin (\frac{π}{8})$ $P = - (sqrt2/2)sin (pi/8)$

Note. We can evaluate P by finding exact value of sin (pi/8), using the trig identity:
$cos (\frac{π}{4}) = \frac{\sqrt{2}}{2} = 1 - 2 {sin}^{2} (\frac{π}{8})$ $cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)$

Answer 2 · 2016-05-21T06:07:17

$sin (\frac{π}{8}) cos (\frac{5 π}{4}) = - \frac{1}{4} \sqrt{4 - 2 \sqrt{2}}$ $sin(pi/8)cos((5pi)/4)=-1/4sqrt(4-2sqrt2)$

Explanation:

$sin (\frac{π}{8}) cos (\frac{5 π}{4})$ $sin(pi/8)cos((5pi)/4)$

Let us first calculate

$cos (\frac{5 π}{4}) = cos (\frac{π}{4} + π) = - cos (\frac{π}{4}) = - \frac{1}{\sqrt{2}}$ $cos((5pi)/4)=cos(pi/4+pi)=-cos(pi/4)=-1/sqrt2$

For $sin (\frac{π}{8})$ $sin(pi/8)$ , let us use the identity $cos 2 θ = 1 - 2 {sin}^{2} θ$ $cos2theta=1-2sin^2theta$

Hence, $cos (\frac{π}{4}) = 1 - 2 {sin}^{2} (\frac{π}{8})$ $cos(pi/4)=1-2sin^2(pi/8)$ or

$\frac{1}{\sqrt{2}} = 1 - 2 {sin}^{2} (\frac{π}{8})$ $1/sqrt2=1-2sin^2(pi/8)$ or

$2 {sin}^{2} (\frac{π}{8}) = 1 - \frac{1}{\sqrt{2}} = 1 - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2}$ $2sin^2(pi/8)=1-1/sqrt2=1-sqrt2/2=(2-sqrt2)/2$

or ${sin}^{2} (\frac{π}{8}) = \frac{2 - \sqrt{2}}{4}$ $sin^2(pi/8)=(2-sqrt2)/4$

or $sin (\frac{π}{8}) = \frac{1}{2} \sqrt{2 - \sqrt{2}}$ $sin(pi/8)=1/2sqrt(2-sqrt2)$

Hence, $sin (\frac{π}{8}) cos (\frac{5 π}{4}) = \frac{1}{2} \sqrt{2 - \sqrt{2}} \times - \frac{\sqrt{2}}{2}$ $sin(pi/8)cos((5pi)/4)=1/2sqrt(2-sqrt2)xx-sqrt2/2$

or $sin (\frac{π}{8}) cos (\frac{5 π}{4}) = - \frac{1}{4} \sqrt{4 - 2 \sqrt{2}}$ $sin(pi/8)cos((5pi)/4)=-1/4sqrt(4-2sqrt2)$

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{4})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 4 )$ without using products of trigonometric functions?

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Explanation:

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{4})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 4 )$ without using products of trigonometric functions?