It is true that $2\sin(54)$ equals $(1+\sqrt5)/2$?

Question

From a popular math site I saw the formula of phi number which is $2 \sin(54^\circ )$. But we also know it is $(1+\sqrt5)/2$ . With calculator this equation has problems. Do you think $2\sin(54)$ is just approximation? By the way is $\sin(54^\circ)$ irrational?

Answer 1

here is how you find it

now you can apply the $sin(3x)$ formula $3*sin(x)-4(sinx)^3$

yes, $sin(54)$ is irrational.

proof:

$sin(54)=(1+\sqrt5)/4$ thus $\sqrt5$=$4*sin(54)-1$

as you know that $\sqrt5$ is irrational

also 1 is rational

thus $4*sin(54)$ has to be irrational

4 is rational and sin(54) is irrational