1-17/42=42/42-17/42=25/42
1-1/100=100/100-1/100=99/100
В первый день прочитано 128/4 = 32 страницы.
<span>А за два дня было прочитано 128-63=65 страниц. </span>
<span>Значит, за один второй день прочитано 65-32=33 страницы </span>
<span>Ответ: 33 страницы.</span>
75/100 + 1/28 + 5/7 = 3/4 + 1/28 + 5/7 = 7* 3/4 + 1/28 + 4* 5/7 =
= 21/28 + 1/28 + 20/28 = 42/28 = 1(целая) 14/28 = 1(целая) 1/2
Воспользуемся формулой:
![\cos{x} + \cos{y} = 2 \cos{ \frac{ x + y }{2} } \cos{ \frac{ x - y }{2} } \ ;](https://tex.z-dn.net/?f=+%5Ccos%7Bx%7D+%2B+%5Ccos%7By%7D+%3D+2+%5Ccos%7B+%5Cfrac%7B+x+%2B+y+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+x+-+y+%7D%7B2%7D+%7D+%5C+%3B+)
И получим, что:
![\cos{3a} + \cos{4a} + \cos{5a} = \cos{3a} + \cos{5a} + \cos{4a} = \\\\ = 2 \cos{ \frac{ 3a + 5a }{2} } \cos{ \frac{ 3a - 5a }{2} } + \cos{4a} = 2 \cos{ \frac{ 8a }{2} } \cos{ \frac{ -2a }{2} } + \cos{4a} = \\\\ = 2 \cos{4a} \cos{a} + \cos{4a} = \cos{4a} ( 2 \cos{a} + 1 ) \ ;](https://tex.z-dn.net/?f=+%5Ccos%7B3a%7D+%2B+%5Ccos%7B4a%7D+%2B+%5Ccos%7B5a%7D+%3D+%5Ccos%7B3a%7D+%2B+%5Ccos%7B5a%7D+%2B+%5Ccos%7B4a%7D+%3D+%5C%5C%5C%5C+%3D+2+%5Ccos%7B+%5Cfrac%7B+3a+%2B+5a+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+3a+-+5a+%7D%7B2%7D+%7D+%2B+%5Ccos%7B4a%7D+%3D+2+%5Ccos%7B+%5Cfrac%7B+8a+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+-2a+%7D%7B2%7D+%7D+%2B+%5Ccos%7B4a%7D+%3D+%5C%5C%5C%5C+%3D+2+%5Ccos%7B4a%7D+%5Ccos%7Ba%7D+%2B+%5Ccos%7B4a%7D+%3D+%5Ccos%7B4a%7D+%28+2+%5Ccos%7Ba%7D+%2B+1+%29+%5C+%3B+)
Вывод:
![\cos{3a} + \cos{4a} + \cos{5a} = \cos{4a} ( 2 \cos{a} + 1 ) \ ;](https://tex.z-dn.net/?f=+%5Ccos%7B3a%7D+%2B+%5Ccos%7B4a%7D+%2B+%5Ccos%7B5a%7D+%3D+%5Ccos%7B4a%7D+%28+2+%5Ccos%7Ba%7D+%2B+1+%29+%5C+%3B+)
![\sin{3a} + \sin{4a} + \sin{5a} = \cos{ ( \frac{ \pi }{2} - 3a ) } + \cos{ ( \frac{ \pi }{2} - 5a ) } + \sin{4a} = \\\\ = 2 \cos{ \frac{ [ \pi/2 - 3a ] + [ \pi/2 - 5a ] }{2} } \cos{ \frac{ [ \pi/2 - 3a ] - [ \pi/2 - 5a ] }{2} } + \sin{4a} = \\\\ = 2 \cos{ \frac{ \pi - 8a }{2} } \cos{ \frac{ 2a }{2} } + \sin{4a} = 2 \cos{ ( \frac{ \pi }{2} - 4a ) } \cos{a} + \sin{4a} \ ;](https://tex.z-dn.net/?f=+%5Csin%7B3a%7D+%2B+%5Csin%7B4a%7D+%2B+%5Csin%7B5a%7D+%3D+%5Ccos%7B+%28+%5Cfrac%7B+%5Cpi+%7D%7B2%7D+-+3a+%29+%7D+%2B+%5Ccos%7B+%28+%5Cfrac%7B+%5Cpi+%7D%7B2%7D+-+5a+%29+%7D+%2B+%5Csin%7B4a%7D+%3D+%5C%5C%5C%5C+%3D+2+%5Ccos%7B+%5Cfrac%7B+%5B+%5Cpi%2F2+-+3a+%5D+%2B+%5B+%5Cpi%2F2+-+5a+%5D+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+%5B+%5Cpi%2F2+-+3a+%5D+-+%5B+%5Cpi%2F2+-+5a+%5D+%7D%7B2%7D+%7D+%2B+%5Csin%7B4a%7D+%3D+%5C%5C%5C%5C+%3D+2+%5Ccos%7B+%5Cfrac%7B+%5Cpi+-+8a+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+2a+%7D%7B2%7D+%7D+%2B+%5Csin%7B4a%7D+%3D+2+%5Ccos%7B+%28+%5Cfrac%7B+%5Cpi+%7D%7B2%7D+-+4a+%29+%7D+%5Ccos%7Ba%7D+%2B+%5Csin%7B4a%7D+%5C+%3B+)
Вывод:
![\sin{3a} + \sin{4a} + \sin{5a} = \sin{4a} ( 2 \cos{a} + 1 ) \ ;](https://tex.z-dn.net/?f=+%5Csin%7B3a%7D+%2B+%5Csin%7B4a%7D+%2B+%5Csin%7B5a%7D+%3D+%5Csin%7B4a%7D+%28+2+%5Ccos%7Ba%7D+%2B+1+%29+%5C+%3B+)
А теперь всё подставляя, получаем, что:
![\frac{ \cos{3a} + \cos{4a} + \cos{5a} }{ \sin{3a} + \sin{4a} + \sin{5a} } = \frac{ \cos{4a} ( 2 \cos{a} + 1 ) }{ \sin{4a} ( 2 \cos{a} + 1 ) } = \frac{ \cos{4a} }{ \sin{4a} } = ctg{4a} \ .](https://tex.z-dn.net/?f=+%5Cfrac%7B+%5Ccos%7B3a%7D+%2B+%5Ccos%7B4a%7D+%2B+%5Ccos%7B5a%7D+%7D%7B+%5Csin%7B3a%7D+%2B+%5Csin%7B4a%7D+%2B+%5Csin%7B5a%7D+%7D+%3D+%5Cfrac%7B+%5Ccos%7B4a%7D+%28+2+%5Ccos%7Ba%7D+%2B+1+%29+%7D%7B+%5Csin%7B4a%7D+%28+2+%5Ccos%7Ba%7D+%2B+1+%29+%7D+%3D+%5Cfrac%7B+%5Ccos%7B4a%7D+%7D%7B+%5Csin%7B4a%7D+%7D+%3D+ctg%7B4a%7D+%5C+.+)