calculate-cos-4-pi-8-cos-4-3pi-8-cos-4-5pi-8-cos-4-7pi-8-

Question Number 48360 by maxmathsup by imad last updated on 22/Nov/18

c a l c u l a t e {c o s}^{4} (\frac{π}{8}) + {c o s}^{4} (\frac{3 π}{8}) + {c o s}^{4} (\frac{5 π}{8}) + {c o s}^{4} (\frac{7 π}{8})

Commented by Abdo msup. last updated on 23/Nov/18

S = {c o s}^{4} (\frac{π}{8}) + {c o s}^{4} (π - \frac{π}{8}) + {c o s}^{4} (\frac{3 π}{8}) + {c o s}^{4} (π - \frac{3 π}{8})

= 2 {c o s}^{4} (\frac{π}{8}) + 2 {c o s}^{4} (\frac{3 π}{8})

= 2 {c o s}^{4} (\frac{π}{8}) + 2 {c o s}^{4} (\frac{π}{2} - \frac{π}{8}) = 2 ({c o s}^{4} (\frac{π}{8}) + {s i n}^{4} (\frac{π}{8}))

= 2 {{({c o s}^{2} \frac{π}{8} + {s i n}^{2} \frac{π}{8})}^{2} - 2 {c o s}^{2} \frac{π}{8} {s i n}^{2} \frac{π}{8}}

= 2 {1 - \frac{1}{2} {s i n}^{2} (\frac{π}{4}) = 2 {1 - \frac{1}{2} \frac{1}{2}} = 2 \frac{3}{4} = \frac{3}{2} .

Answered by behi83417@gmail.com last updated on 22/Nov/18

c^{4} (\frac{π}{8}) = c^{4} (\frac{7 π}{8}) = {(\frac{1 + c \frac{π}{4}}{2})}^{2} = {(\frac{1 + \frac{\sqrt{2}}{2}}{2})}^{2} = \frac{3 + 2 \sqrt{2}}{8}

c^{4} (\frac{3 π}{8}) = c^{4} (\frac{5 π}{8}) = {(\frac{1 + c \frac{3 π}{4}}{2})}^{2} = \frac{3 - 2 \sqrt{2}}{8}

\Rightarrow S = 2 \times \frac{3 + 2 \sqrt{2}}{8} + 2 \times \frac{3 - 2 \sqrt{2}}{8} = \frac{3}{2} .

Answered by hknkrc46 last updated on 24/Dec/18

α + β = π

\Rightarrow \cos^{2 k} α = \cos^{2 k} β (k \in N)

α = \frac{π}{8}, β = \frac{7 π}{8}, k = 2

\Rightarrow \cos^{4} (\frac{π}{8}) = \cos^{4} (\frac{7 π}{8})

\Rightarrow \cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{7 π}{8}) = 2 \cos^{4} (\frac{π}{8})

α = \frac{3 π}{8}, β = \frac{5 π}{8}, k = 2

\Rightarrow \cos^{4} (\frac{3 π}{8}) = \cos^{4} (\frac{5 π}{8})

\Rightarrow \cos^{4} (\frac{3 π}{8}) + \cos^{4} (\frac{5 π}{8}) = 2 \cos^{4} (\frac{3 π}{8})

2 \cos^{4} (\frac{π}{8}) + 2 \cos^{4} (\frac{3 π}{8})

= 2 [\cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{3 π}{8})]

= 2 [\cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{π}{2} - \frac{π}{8})]

= 2 [\cos^{4} (\frac{π}{8}) + \sin^{4} (\frac{π}{8})]

= 2 [{(\cos^{2} (\frac{π}{8}) + \sin^{2} (\frac{π}{8}))}^{2} - \frac{4 \cos^{2} (\frac{π}{8}) \sin^{2} (\frac{π}{8})}{2}]

= 2 [1^{2} - \frac{{(2 \cos (\frac{π}{8}) \sin (\frac{π}{8}))}^{2}}{2}]

= 2 [1 - \frac{\sin^{2} (\frac{π}{4})}{2}] = 2 - \sin^{2} (\frac{π}{4})

= 2 - {(\frac{\sqrt{2}}{2})}^{2} = 2 - \frac{1}{2} = \frac{3}{2}

Facebook Tweet Pin