Question

How am I supposed to do this without a calculator?

Answer 1

Answered by Mimiwhatsup:

`\mathrm{Use\:the\:following\:identity}:\quad \cos \left(s\right)\cos \left(t\right)+\sin \left(s\right)\sin \left(t\right)=\cos \left(s-t\right)\\\sin \left((2\pi )/(9)\right)\sin \left((\pi )/(18)\right)+\cos \left((2\pi )/(9)\right)\cos \left((\pi )/(18)\right)=\cos \left((2\pi )/(9)-(\pi )/(18)\right)\\=\cos \left((2\pi )/(9)-(\pi )/(18)\right)\\\mathrm{Join}\:(2\pi )/(9)-(\pi )/(18):\quad (\pi )/(6)\\(2\pi )/(9)-(\pi )/(18)\\`

`\mathrm{Least\:Common\:Multiplier\:of\:}9,\:18:\quad 18\\Adjust\:Fractions\:based\:on\:the\:LCM\\=(4\pi )/(18)-(\pi )/(18)\\\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\quad (a)/(c)\pm (b)/(c)=(a\pm \:b)/(c)\\=(4\pi -\pi )/(18)\\4\pi -\pi =3\pi \\4\pi -\pi =3\pi \\\mathrm{Cancel\:the\:common\:factor:}\:3\\=(\pi )/(6)\\=\cos \left((\pi )/(6)\right)\\`

`Answer: \mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left((\pi )/(6)\right)=(√(3))/(2)`