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العربية

cos(pi/5)cos((2pi)/5)cos((4pi)/5)cos((8pi)/5)

Solution

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

Décimale

- 0.0625

étapes des solutions

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

Récrire en utilisant des identités trigonométriques:

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

Utiliser l'identité du produit à la somme:

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (\frac{4 π}{5} - \frac{8 π}{5}) + cos (\frac{4 π}{5} + \frac{8 π}{5}))

Simplifier:

\frac{4 π}{5} - \frac{8 π}{5} = - \frac{4 π}{5}

\frac{4 π}{5} - \frac{8 π}{5}

Appliquer la règle

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{4 π - 8 π}{5}

Additionner les éléments similaires :

4 π - 8 π = - 4 π

= \frac{- 4 π}{5}

Appliquer la règle des fractions:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 π}{5}

Simplifier:

\frac{4 π}{5} + \frac{8 π}{5} = \frac{12 π}{5}

\frac{4 π}{5} + \frac{8 π}{5}

Appliquer la règle

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{4 π + 8 π}{5}

Additionner les éléments similaires :

4 π + 8 π = 12 π

= \frac{12 π}{5}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (- \frac{4 π}{5}) + cos (\frac{12 π}{5})) = \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (- \frac{4 π}{5}) + cos (\frac{12 π}{5}))

Multiplier des fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

Multiplier:

1 \cdot cos (\frac{π}{5}) = cos (\frac{π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

Utiliser la propriété suivante :

cos (- x) = cos (x)

cos (- \frac{4 π}{5}) = cos (\frac{4 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

cos (\frac{12 π}{5}) = cos (\frac{2 π}{5})

cos (\frac{12 π}{5})

Récrire

\frac{12 π}{5}

comme

2 π + \frac{2 π}{5}

= cos (2 π + \frac{2 π}{5})

Appliquer la périodicité de

cos

cos (x + 2 π) = cos (x)

cos (2 π + \frac{2 π}{5}) = cos (\frac{2 π}{5})

= cos (\frac{2 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2} = \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

Développer

cos (\frac{π}{5}) cos (\frac{2 π}{5}) (cos (\frac{4 π}{5}) + cos (\frac{2 π}{5})) : cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) (cos (\frac{4 π}{5}) + cos (\frac{2 π}{5}))

Appliquer la loi de la distribution:

a (b + c) = ab + a c

a = cos (\frac{π}{5}) cos (\frac{2 π}{5}), b = cos (\frac{4 π}{5}), c = cos (\frac{2 π}{5})

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5}) = cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5})

Appliquer la règle de l'exposant:

a^{b} \cdot a^{c} = a^{b + c}

cos (\frac{2 π}{5}) cos (\frac{2 π}{5}) = cos^{1 + 1} (\frac{2 π}{5})

= cos (\frac{π}{5}) cos^{1 + 1} (\frac{2 π}{5})

Additionner les nombres :

1 + 1 = 2

= cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

sin (\frac{π}{2} - 1)

tan^{2} (2 7^{\circ})

arctan (\frac{36}{48})

\frac{10}{cos ( 4 0 ^{\circ} )}

(tan (4 5^{\circ}) + tan (6 0^{\circ}))^{2}