Answer to Question #312752 in Algebra for deepu

Question

prove that sin (degree(1)) is an algebraic number

Answer 1

180^o= $\pi$ radian

Let 1 be k

sin k^o

k^o= k$\pi$/180

e^k$\pi$^i/180 = cos ( k$\pi$/180)+ isin (k$\pi$ /180)

(e^k$\pi$^i/180)¹⁸⁰ = cos (k$\pi$) + i sin (k$\pi$)

(-1)^k =+1= -1

cos (k$\pi$/180) + i sin (k$\pi$/180)

cos (k$\pi$/180) - i sin (k$\pi$/180)

2cos (k$\pi$/180) is an algebraic integer

cos (k$\pi$/180) becomes an algebraic integer

i sin (k$\pi$/180) is an algebraic integer

sin (k$\pi$/180) is also an algebraic integer

$\therefore$ sin (degree(1)) is an algebraic number