cos-pi-7-cos-2pi-7-cos-3pi-7-

Question Number 112645 by bemath last updated on 09/Sep/20

\cos (\frac{π}{7}) - \cos (\frac{2 π}{7}) + \cos (\frac{3 π}{7}) = ?

Answered by john santu last updated on 09/Sep/20

l e t p = \frac{π}{7} a n d X = \cos p - \cos 2 p + \cos 3 p

m u l t i p l y b y 2 \sin p

2 \sin p X = 2 \sin p (\cos p - \cos 2 p + \cos 3 p)

2 \sin p X = \sin 2 p - 2 \sin p \cos 2 p + 2 \sin p \cos 3 p

2 \sin p X = \sin 2 p - (\sin 3 p - \sin p) + \sin 4 p - \sin 2 p

2 \sin p X = \sin 4 p + \sin p - \sin 3 p

2 \sin p X = 2 \cos (\frac{7 p}{2}) \sin (\frac{p}{2}) + \sin p

c o n s i d e r \cos (\frac{7 p}{2}) = \cos (\frac{7}{2} . \frac{π}{7}) = 0

s o w e g e t 2 \sin p X = \sin p

X = \frac{1}{2} .

Answered by 1549442205PVT last updated on 09/Sep/20

P = \cos (\frac{π}{7}) - \cos (\frac{2 π}{7}) + \cos (\frac{3 π}{7})

= \cos \frac{π}{7} - ({2 \cos}^{2} \frac{π}{7} - 1) + {4 \cos}^{3} \frac{π}{7} - 3 \cos \frac{π}{7}

= {4 \cos}^{3} \frac{π}{7} - {2 \cos}^{2} \frac{π}{7} - 2 \cos \frac{π}{7} + 1 (1)

On the other hands, we have

{4 \cos}^{3} \frac{π}{7} - 3 \cos \frac{π}{7} = \cos \frac{3 π}{7} = - \cos (π - \frac{3 π}{7})

= - \cos \frac{4 π}{7} = 1 - {2 \cos}^{2} \frac{2 π}{7} = 1 - 2 {({2 \cos}^{2} \frac{π}{7} - 1)}^{2}

= 1 - 2 ({4 \cos}^{4} \frac{π}{7} - {4 \cos}^{2} \frac{π}{7} + 1)

= - {8 \cos}^{4} \frac{π}{7} + {8 \cos}^{2} \frac{π}{7} - 1 . Hence, we get

{8 \cos}^{4} \frac{π}{7} + {4 \cos}^{3} \frac{π}{7} - {8 \cos}^{2} \frac{π}{7} - 3 \cos \frac{π}{7} + 1 = 0

\Leftrightarrow (\cos \frac{π}{7} + 1) ({8 \cos}^{3} \frac{π}{7} - {4 \cos}^{2} \frac{π}{7} - 4 \cos \frac{π}{7} + 1) = 0

\Leftrightarrow {8 \cos}^{3} \frac{π}{7} - {4 \cos}^{2} \frac{π}{7} - 4 \cos \frac{π}{7} + 1 = 0

\Rightarrow {4 \cos}^{3} \frac{π}{7} - {2 \cos}^{2} \frac{π}{7} - 2 \cos \frac{π}{7} + \frac{1}{2} = 0 (2)

From (1) and (2) we get

P = ({4 \cos}^{3} \frac{π}{7} - {2 \cos}^{2} \frac{π}{7} - 2 \cos \frac{π}{7} + \frac{1}{2}) + \frac{1}{2}

= 0 + \frac{1}{2} . \boldsymbol Thus, \boldsymbol P = \boldsymbol \cos (\frac{\boldsymbol π}{7}) - \boldsymbol \cos (\frac{2 \boldsymbol π}{7}) + \boldsymbol \cos (\frac{3 \boldsymbol π}{7}) = \frac{1}{2}