Find-the-value-of-sin-pi-9-in-exact-form-

Question Number 123248 by Dwaipayan Shikari last updated on 24/Nov/20

F i n d t h e v a l u e o f s i n (\frac{π}{9}) i n e x a c t f o r m

Commented by MJS_new last updated on 24/Nov/20

\sin \frac{π}{9} = \sin 20 °

I {don}^{'} t think {it}^{'} s possible . the smallest

angle α for which we can exactly express

\sin α is \frac{π}{60} = 3 ° \Rightarrow for β with β \neq \frac{n π}{60} or β \neq (3 n) °

this is impossible . I saw an “ {expression}^{″} for

\sin 1 ° but it uses {Cardano}^{'} s Solution for

x^{3} + p x + q = 0 while \frac{p^{3}}{27} + \frac{q^{2}}{4} < 0 but in this case

{it}^{'} s not defined . you can write it down but it

makes no sense at all .

Commented by Dwaipayan Shikari last updated on 24/Nov/20

s i n (\frac{π}{9}) = s i n θ

9 θ = π \Rightarrow s i n 6 θ = s i n 3 θ \Rightarrow 2 c o s 3 θ = 1 (s i n θ = \sqrt{1 - t^{2}}

4 t^{3} - 3 t = \frac{1}{2} \Rightarrow t^{3} - \frac{3}{4} t - \frac{1}{8} = 0

\frac{1}{27} {(\frac{- 3}{4})}^{3} + \frac{1}{4} {(- \frac{1}{8})}^{2} = - \frac{1}{64} + \frac{1}{4.64} < 0 (I n v a l i d)

B u t a n y o t h e r w a y s i r ?

Commented by MJS_new last updated on 24/Nov/20

no other way

Answered by zarminaawan last updated on 24/Nov/20

s i n 20 = 0.342

Commented by MJS_new last updated on 24/Nov/20

the question is “ \dots i n e x a c t {f o r m}^{″}

Answered by TANMAY PANACEA last updated on 24/Nov/20

y = s i n x

y + △ y = s i n (x + △ x)

h e r e △ x = 2^{o} = (\frac{π}{180} \times 2) = \frac{π}{90}

x = 18^{o}

\frac{△ y}{△ x} \approx \frac{d y}{d x}

△ y = \frac{d y}{d x} △ x

△ y = c o s x \times (\frac{π}{90})

s i n (20^{o}) = s i n 18^{o} + \frac{π}{90} \times c o s 18^{o}

= \frac{\sqrt{5} - 1}{4} + (\frac{π}{90}) \times \sqrt{\frac{10 + 2 \sqrt{5}}{4}}

= 0.30901699437 + \frac{π}{90} \times 0.9510565163 (u s i n g c a l c u l a t o r)

= 0.30901699437 + 0.03321149739

= 0.34222849176

Commented by MJS_new last updated on 24/Nov/20

interesting!

but {it}^{'} s also no exact solution \dots I {don}^{'} t think

{it}^{'} s possible

Commented by TANMAY PANACEA last updated on 24/Nov/20

t h a n k y o u s i r . . y e s y o u c o r r e c t

Commented by Dwaipayan Shikari last updated on 24/Nov/20

T h a n k i n g f o r i n t e r a c t i o n s i r s!