Find sin72 if you know the sines of 30 45 60?

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Find sin72 if you know the sines of 30 45 60?

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Answer 1 · 2018-07-20T16:45:34

The answer is $= \frac{\sqrt{10 + 2 \sqrt{5}}}{4}$ $=sqrt(10+2sqrt5)/4$

Explanation:

We need

$sin (2 x) = 2 sin x cos x$ $sin(2x)=2sinxcosx$

$cos (2 x) = 1 - 2 {sin}^{2} x$ $cos(2x)=1-2sin^2x$

$sin x = cos (90 - x)$ $sinx=cos(90-x)$

${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$

Let's start

$sin (72) = sin (2 \cdot 36)$ $sin(72)=sin(2*36)$

$= 2 sin 36 cos 36$ $=2sin36cos36$

$= 2 sin (2 \cdot 18) cos (2 \cdot 18)$ $=2sin(2*18)cos(2*18)$

$= 2 sin (2 sin 18 cos 18) (1 - 2 {sin}^{2} 18)$ $=2sin(2sin18cos18)(1-2sin^2 18)$

$= 4 sin 18 cos 18 (1 - 2 {sin}^{2} 18)$ $=4sin18cos18(1-2sin^2 18)$

Therefore,

$sin 72 = cos (90 - 72) = 4 sin 18 cos 18 (1 - 2 {sin}^{2} 18)$ $sin72=cos(90-72)=4sin18cos18(1-2sin^2 18)$

$cos 18 = 4 sin 18 cos 18 (1 - 2 {sin}^{2} 18)$ $cos18=4sin18cos18(1-2sin^2 18)$

Divide both sides by $cos 18$ $cos18$

$1 = 4 sin 18 (1 - 2 {sin}^{2} 18)$ $1=4sin18(1-2sin^2 18)$

$8 {sin}^{3} 18 - 4 sin 18 + 1 = 0$ $8sin^3 18-4sin18+1=0$

Factorise

$(2 sin 18 - 1) (4 {sin}^{2} 18 + 2 sin 18 - 1) = 0$ $(2sin18-1)(4sin^2 18+ 2sin18-1)=0$

Therefore,

$sin 18 = \frac{1}{2}$ $sin18=1/2$ which is $\emptyset$ $O/$

The solution to the quadratic equation is

$sin 18 = \frac{- 2 \pm \sqrt{2^{2} - 4 \cdot 4 \cdot (- 1)}}{2 \cdot 4}$ $sin18=(-2+-sqrt(2^2-4*4*(-1)))/(2*4)$

$= \frac{- 2 \pm \sqrt{20}}{8}$ $=(-2+-sqrt20)/8$

$= \frac{- 2 \pm 2 \sqrt{5}}{8}$ $=(-2+-2sqrt5)/8$

$= \frac{- 1 \pm \sqrt{5}}{4}$ $=(-1+-sqrt5)/4$

Discard $sin 18 = \frac{- 1 - \sqrt{5}}{4}$ $sin18=(-1-sqrt5)/4$ as this is $< 0$ $<0$

We keep only $sin 18 = \frac{- 1 + \sqrt{5}}{4}$ $sin18=(-1+sqrt5)/4$

So,

$cos 72 = \frac{- 1 + \sqrt{5}}{4}$ $cos72=(-1+sqrt5)/4$

$sin 72 = \sqrt{1 - {cos}^{2} 72}$ $sin72=sqrt(1-cos^2 72)$

$= \sqrt{1 - {(\frac{- 1 + \sqrt{5}}{4})}^{2}}$ $=sqrt(1-((-1+sqrt5)/4)^2)$

$= \frac{\sqrt{(16 - (1 + 5 - 2 \sqrt{5}))}}{4}$ $=sqrt((16-(1+5-2sqrt5)))/4$

$= \frac{\sqrt{10 + 2 \sqrt{5}}}{4}$ $=sqrt(10+2sqrt5)/4$

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Find sin72 if you know the sines of 30 45 60?

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