find-cos-5-interms-of-cos-then-find-the-value-of-cos-pi-10-

Question Number 29173 by abdo imad last updated on 04/Feb/18

f i n d c o s (5 α) i n t e r m s o f c o s α t h e n f i n d t h e v a l u e o f

c o s (\frac{π}{10}) .

Answered by ajfour last updated on 05/Feb/18

\cos 5 α = \cos 3 α \cos 2 α - \sin 3 α \sin 2 α

= (4 \cos^{3} α - 3 \cos α) (2 \cos^{2} α - 1)

- (3 \sin α - 4 \sin^{3} α) (2 \sin α \cos α)

= \cos α (4 \cos^{2} α - 3) (2 \cos^{2} α - 1)

- 2 \cos α (1 - \cos^{2} α) (4 \cos^{2} α - 1)

= \cos α (8 \cos^{4} α - 10 \cos^{2} α + 3

+ 8 \cos^{4} α - 10 \cos^{2} α + 2)

= \cos α (16 \cos^{4} α - 20 \cos^{2} α + 5)

l e t α = \frac{π}{2} a n d \cos^{2} α = t

\Rightarrow 16 t^{2} - 20 t + 5 = 0

16 {(t - \frac{5}{8})}^{2} = \frac{25}{4} - 5

t = \cos^{2} α = \frac{5}{8} + \frac{\sqrt{5}}{8}

\Rightarrow \cos α = \cos \frac{π}{10} = \sqrt{\frac{5 + \sqrt{5}}{8}} .

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