Question

Найдите синус , косинус, тангенс, котангенс 22*30'

Answer 1

$22^{circ} 30' = dfrac{180^{circ}}{8} = dfrac{45^{circ}}{2} \ \ sin dfrac{45^{circ}}{2} = sqrt{ dfrac{1 - cos45^{circ}}{2} } = sqrt{ dfrac{1 - dfrac{ sqrt{2} }{2} }{2} } = sqrt{ dfrac{ dfrac{ 2 - sqrt{2 } }{2} }{2} } = dfrac{ 2 - sqrt{2} }{2} \ \ cos dfrac{45^{circ}}{2} = sqrt{ dfrac{1 +cos45^{circ}}{2} } = sqrt{ dfrac{1 + dfrac{ sqrt{2} }{2} }{2} } = sqrt{ dfrac{ dfrac{ 2 + sqrt{2 } }{2} }{2} } = dfrac{ 2 + sqrt{2} }{2}$
$tg dfrac{45^{circ}}{2} = dfrac{1 - cos45^{circ}}{sin45^{circ}} = dfrac{1 - dfrac{ sqrt{2} }{2} }{ dfrac{ sqrt{2} }{2} } = sqrt{2} - 1 \ ctg dfrac{45^{circ}}{2} = dfrac{1}{tg dfrac{45^{circ}}{2} } = dfrac{1}{ sqrt{2} -1} = dfrac{1 + sqrt{2} }{2 - 1} = 1 + sqrt{2}$