Menghitung Nilai Sinus dan Cosinus 36 Derajat tanpa Kalkulator

Rumus Dasar Trigonmetri yang dibutuhkan

$ \spadesuit \, $ Rumus trigonometri sudut ganda
$ \cos 2A = 1 - 2\sin ^2 A $

$ \clubsuit \, $ Identitas trigonmetri
$ \sin ^2 A + \cos ^2 A = 1 \rightarrow \sin A = \sqrt{1 - \cos ^2 A} $

Kita membutuhkan nilai sin 18 derajat :
$ \sin 18^\circ = \frac{-1 + \sqrt{5}}{4} $

Nilai cos 36 derajat dan sin 36 derajat

$ \cos 36^\circ = \frac{1}{4}(1 + \sqrt{5} ) $
$ \sin 36^\circ = \frac{1}{4} \sqrt{ 10 - 2\sqrt{5} } $

Mencari nilai cos 36 derajat dan sin 36 derajat
*). Menentukan nilai cos 36 derajat
$ \begin{align} \cos 2A & = 1 - 2\sin ^2 A \\ \cos 36^\circ & = 1 - 2\sin ^2 18^\circ \\ & = 1 - 2 (\frac{-1 + \sqrt{5}}{4})^2 \\ & = 1 - 2 \times \frac{6 - 2 \sqrt{5}}{16} \\ & = 1 - 2 \times 2. \frac{3 - \sqrt{5}}{16} \\ & = 1 - \frac{3 - \sqrt{5}}{4} \\ & = \frac{4}{4} - \frac{3 - \sqrt{5}}{4} \\ & = \frac{4 - (3 - \sqrt{5}) }{4} \\ & = \frac{1 + \sqrt{5} }{4} \\ & = \frac{1}{4}(1 + \sqrt{5} ) \end{align} $
Jadi, nilai $ \cos 36^\circ = \frac{1}{4}(1 + \sqrt{5} ) $

*). Menentukan nilai sin 36 derajat :
$ \begin{align} \cos 36^\circ & = \frac{1}{4}(1 + \sqrt{5} ) \, \, \, \, \, \text{(kuadratkan)} \\ \cos ^2 36^\circ & = [ \frac{1}{4}(1 + \sqrt{5} ) ]^2 \\ & = \frac{1}{16}(6 + 2\sqrt{5} ) \, \, \, \, \, \text{(identitas)} \\ \sin A & = \sqrt{1 - \cos ^2 A} \\ \sin 36^\circ & = \sqrt{1 - \cos ^2 36^\circ } \\ & = \sqrt{1 - \frac{1}{16}(6 + 2\sqrt{5} ) } \\ & = \sqrt{\frac{16}{16} - \frac{1}{16}(6 + 2\sqrt{5} ) } \\ & = \sqrt{ \frac{1}{16}(16- (6 + 2\sqrt{5} ) ) } \\ & = \sqrt{ \frac{1}{16}(10 - 2\sqrt{5} )} \\ & = \frac{1}{4} \sqrt{ 10 - 2\sqrt{5} } \end{align} $
Jadi, nilai $ \sin 36^\circ = \frac{1}{4} \sqrt{ 10 - 2\sqrt{5} } $