Question

How do you solve cos(pi/24) using Half-Angle formulas, and leaving in simplified form?

Answer 1

`\cos (\theta)/(2)=\sqrt{(1+\cos \theta)/(2)} \\ \\ \cos{(\pi)/(24)}=\cos{((\pi)/(12))/(2)}=\sqrt{(1+\cos (\pi)/(12))/(2)}= \sqrt{ (1)/(2)+ (\cos (\pi)/(12))/(2) } \\ \\ \cos{(\pi)/(12)}=\cos{((\pi)/(6))/(2)}=\sqrt{(1+\cos (\pi)/(6))/(2)}= \sqrt{ (1)/(2)+ ( ( √(3) )/(2) )/(2) } =\sqrt{ (2)/(4)+ ( √(3) )/(4) } = \sqrt{( 2+√(3) )/(4) } = \\ \\ =\frac{ \sqrt{2+√(3)} }{2}`


`\cos{(\pi)/(24)}= \sqrt{ (1)/(2)+ (\cos (\pi)/(12))/(2) } = \sqrt{ (1)/(2)+ \frac{\frac{ \sqrt{2+√(3)} }{2}}{2} } = \sqrt{ (2)/(4)+ \frac{ \sqrt{2+√(3)} }{4}} } =\sqrt{ \frac{ 2+\sqrt{2+√(3)} }{4}} } \\ \\\cos{(\pi)/(24)}=\frac{\sqrt{2+\sqrt{2+√(3)}}} {2}`