How do I evaluate cos(pi/5) without using a calculator?

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How do I evaluate cos(pi/5) without using a calculator?

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Answer 1 · 2015-09-08T04:11:38

Cos ( $π$ $pi$ /5) = cos 36° = ( $\sqrt{}$ $sqrt$ 5 + 1)/4.

Explanation:

If $θ$ $theta$ = $π$ $pi$ /10, then 5 $θ$ $theta$ = $π$ $pi$ /2 $\Rightarrow$ $=>$ cos3 $θ$ $theta$ = sin2 $θ$ $theta$ .[ cos ( $π$ $pi$ /2 - $α$ $alpha$ ) = sin $α$ $alpha$ }.
$\Rightarrow$ $=>$ 4 ${cos}^{3}$ $cos ^3$ $θ$ $theta$ - 3cos $θ$ $theta$ = 2sin $θ$ $theta$ cos $θ$ $theta$ $\Rightarrow$ $=>$ 4 ${cos}^{2}$ $cos^2$ $θ$ $theta$ - 3 =2 sin $θ$ $theta$ .
$\Rightarrow$ $=>$ 4 ( 1 - ${sin}^{2}$ $sin^2$ $θ$ $theta$ ) - 3 = 2 sin $θ$ $theta$ . $\Rightarrow$ $=>$ 4 ${sin}^{2}$ $sin^2$ $θ$ $theta$ +2sin $θ$ $theta$ - 1 = 0 $\Rightarrow$ $=>$
sin $θ$ $theta$ =( $\sqrt{}$ $sqrt$ 5 - 1 ) /4.
Now cos 2 $θ$ $theta$ = cos $π$ $pi$ /5 = 1 - 2 ${sin}^{2}$ $sin^2$ $θ$ $theta$ , gives the result.

Answer 2 · 2016-02-13T16:21:40

$cos (\frac{π}{5}) = \frac{\sqrt{5} + 1}{4}$ $Cos (pi/5) = (sqrt (5)+1)/4$ .

Explanation:

Let $a = cos (\frac{π}{5})$ $a = cos(pi/5)$ , $b = cos (2 \cdot \frac{π}{5})$ $b = cos(2*pi/5)$ . Thus $cos (4 \cdot \frac{π}{5}) = - a$ $cos(4*pi/5) = -a$ . From the double angle formulas:

$b = 2 a^{2} - 1$ $b = 2a^2-1$
$- a = 2 b^{2} - 1$ $-a = 2b^2-1$

Subtracting,

$a + b = 2 (a^{2} - b^{2}) = 2 (a + b) (a - b)$ $a+b = 2(a^2-b^2) = 2(a+b)(a-b)$

$a + b$ $a+b$ is not zero, as both terms are positive, so $a - b$ $a-b$ must be $\frac{1}{2}$ $1/2$ . Then

$a - \frac{1}{2} = 2 a^{2} - 1$ $a-1/2 = 2a^2-1$
$4 a^{2} - 2 a - 1 = 0$ $4a^2-2a-1 = 0$

and the only positive root is

$a = cos (\frac{π}{5}) = \frac{\sqrt{5} + 1}{4}$ $a = cos (pi/5) = (sqrt(5)+1)/4$ .

And $b = cos (2 \cdot \frac{π}{5}) = a - \frac{1}{2} = \frac{\sqrt{5} - 1}{4}$ $b = cos (2*pi/5) = a-1/2 = (sqrt(5)-1)/4$ .

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How do I evaluate cos(pi/5) without using a calculator?

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