The value of cos 36° could be determined by golden triangle, which is

\cos 36\degree = \dfrac{1+\sqrt{5} }{4}

Using Pythagorean Theorem,

\sin 36° = \sqrt{1-\cos^2 36\degree }

=\sqrt{1-( \dfrac{1+\sqrt{5} }{4})^2}

=\dfrac{1}{4}\cdotp \sqrt{16-(6-2\sqrt{5} )}

=\dfrac{1}{4}\cdotp \sqrt{10-2\sqrt{5} }

Therefore, the exact value of sin 36° is \dfrac{1}{4}\cdotp \sqrt{10-2\sqrt{5} }, which is approximately equal to 0.587785252

Use the following Excel formula to verify,

=SQRT(10-2*SQRT(5))/4