Given that cos 30 = 3 / 2 \cos 30 = \sqrt{3}/2 cos 30 = 3 /2 and that cos 36 = ( 1 + 5 ) / 4 \cos 36 = (1 + \sqrt{5})/4 cos 36 = ( 1 + 5 ) /4 , find (using formulas for the cosines of the sums and differences of angles) an exact expression for cos 3 \cos 3 cos 3 . Your expression may use radicals as they are exact.
Solution We will need sin 30 \sin 30 sin 30 and sin 36 \sin 36 sin 36 as well as the given cosines. These follow from the Pythagorean formula: ( cos θ ) 2 + ( sin θ ) 2 = 1 (\cos \theta)^2 + (\sin \theta)^2 = 1 ( cos θ ) 2 + ( sin θ ) 2 = 1 . Using this we find that
sin 30 = 1 − ( cos 30 ) 2 = 1 − 3 4 = 1 2 sin 36 = 1 − ( cos 36 ) 2 = 1 − 1 + 5 + 2 5 16 = 16 − 6 − 2 5 16 = 10 − 2 5 4 .
\begin{aligned}
\sin 30 &=& \sqrt{1 - (\cos 30)^2} = \sqrt{1 - \frac{3}{4}} = \frac{1}{2} \\
\sin 36 &=& \sqrt{1 - (\cos 36)^2} = \sqrt{1 - \frac{1 + 5 + 2 \sqrt{5}}{16}} \\
&=& \sqrt{\frac{16 - 6 - 2 \sqrt{5}}{16}} = \frac{\sqrt{10 - 2 \sqrt{5}}}{4}.
\end{aligned}
sin 30 sin 36 = = = 1 − ( cos 30 ) 2 = 1 − 4 3 = 2 1 1 − ( cos 36 ) 2 = 1 − 16 1 + 5 + 2 5 16 16 − 6 − 2 5 = 4 10 − 2 5 .
After some thought and a bit of scratch work, we find that
cos 3 = cos 6 2 = cos ( 36 − 30 2 ) cos ( 36 − 30 ) = cos 36 cos 30 − sin 36 sin 30 = 1 + 5 4 ⋅ 3 2 − 10 − 2 5 4 ⋅ 1 2 = 3 + 5 + 10 − 2 5 8 cos 3 = cos ( 6 2 ) = 1 + cos 6 2 = 8 + 3 + 15 + 10 − 2 5 16 = 8 + 3 + 15 + 10 − 2 5 4
\begin{aligned}
\cos 3 &=& \cos \frac{6}{2} = \cos \left( \frac{36 - 30}{2} \right) \\
\cos (36 - 30) &=& \cos 36 \cos 30 - \sin 36 \sin 30 \\
&=& \frac{1+\sqrt{5}}{4} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{10 - 2 \sqrt{5}}}{4} \cdot \frac{1}{2} \\
&=& \frac{\sqrt{3} + \sqrt{5} + \sqrt{10 - 2 \sqrt{5}}}{8} \\
\cos 3 &=& \cos \left( \frac{6}{2} \right) = \sqrt{ \frac{1 + \cos{6}}{2}} \\
&=& \sqrt{ \frac{8 + \sqrt{3} + \sqrt{15} + \sqrt{10 - 2 \sqrt{5}}}{16}} \\
&=& \frac{ \sqrt{8 + \sqrt{3} + \sqrt{15} + \sqrt{10 - 2 \sqrt{5}}}}{4}
\end{aligned}
cos 3 cos ( 36 − 30 ) cos 3 = = = = = = = cos 2 6 = cos ( 2 36 − 30 ) cos 36 cos 30 − sin 36 sin 30 4 1 + 5 ⋅ 2 3 − 4 10 − 2 5 ⋅ 2 1 8 3 + 5 + 10 − 2 5 cos ( 2 6 ) = 2 1 + cos 6 16 8 + 3 + 15 + 10 − 2 5 4 8 + 3 + 15 + 10 − 2 5
Wow. Complicated!