# rarrtan75 ° = tan (45 + 30) #
# = (tan45 + tan30) / (1-tan45 * tan30) #
# 1 (1+ (1 / sqrt (3))) / (1- (1 / sqrt (3)) #
# = (sq) (3) 1) / (sqrt (3) -1) = 2 + sqrt (3) #
# rarrtan52.5 = מיטה (90-37.5) = cot37.5 #
# rarrcot37.5 = 1 / (tan (75/2)) #
# rarrtanx = (2tan (x / 2)) / (1-tan ^ 2 (x / 2)) #
# rarrtanx-tanx * tan ^ 2 (x / 2) = 2tan (x / 2) # #
# rarrtanx * tan ^ 2 (x / 2) + 2tan (x / 2) -tanx = 0 #
זה ריבועי פנימה #tan (x / 2) # # לכן, # (2/2) * tanx * (- tanx)) / (2 * tanx) # #
# (2/2) = (+ 2) + (2) + (2 + tanx)
#rarrtan (x / 2) = (- 1 + sqrt (1 + tan ^ 2x)) / tanx #
לשים # x = 75 # אנחנו מקבלים
# (1) = (1) + (1) + (1) + tan ^ 2 (75)) (/ tan75) #
# (1) (1) (1 +) 2 + sqrt (3)) ^ 2)) / (2 + sqrt (3)) #
# (1) 4 + 4sqrt (3) +3)) / (2 + sqrt (3)) #
# (1) + (2 + 4sqrt (3))) / (2 + sqrt (3)) #
# 2) (2) * (2) sqrt (2 + sqrt) (1) + 1) (2 / sqrt (3)) / (2 * sqrt (2 + sqrt) / (2 * sqrt (2 + sqrt (3) +1) #
# (2) * 2 * sqrt (2 + sqrt (3)) + 1)) / () 2 (* sqrt (2 + sqrt (3))) ^ 2-1 ^ 2) #
תן #sqrt (2 + sqrt (3)) = a #
# (2 * a + 1)) (2 * 2 * a + 1) + 3) * (2 * a + 1)
# # rarrcot37.5 = (4a + 2 + 2sqrt (3) a + sqrt (3)) / ((4 * (2 + sqrt (3)) - 1 #
# 4 rccot37.5 = (4a + 2sqrt (3) a + a ^ 2) / (7 + 4sqrt (3)) (7-4sqrt (3)) / (7-4sqrt (3)) #
# rarrcot37.5 = 7 * (4a + 2sqrt (3) a + a ^ 2) -4sqrt (3) * (4a + 2sqrt (3) a + a ^ 2) #
# rarrcot37.5 = 28a + 14sqrt (3) a + 7a ^ 2-16sqrt (3) a-24a-4sqrt (3) a ^ 2 #
# rarrcot37.5 = 7a ^ 2-4sqrt (3) a ^ 2 + 4a-2sqrt (3) a #
# rarrcot37.5 = a ^ 2 (7-4sqrt (3)) + 2 * a (2-sqrt (3)) #
# sqrrt (2 + sqrt (3)) * sqrt (2-sqrt (3)) * sqrt (2) -qqrt (3))) #
# # rarrcot37.5 = 2 * (2 * 2 * 2 * 2) * 2 (2) *
# rarrcot37.5 = 14-8sqrt (3) + 7sqrt (3) -12 + sqrt (sqrt (6) -sqrt (2)) ^ 2) # #
# rarrtan52.5 = 2-sqrt (3) + sqrt (6) -qqrt (2) #
הוכיח.
תשובה:
גישה קטנה יותר …
כללים משומשים: -
#color (אדום) (ul (bar (צבע (ירוק) (sin2theta = 2 cdot sintheta cdot costheta)) #
# cos2theta = 2cos ^ 2theta-1 #
# => צבע (אדום) (ul (bar (צבע (כחול) (2cos ^ 2theta = 1 + cos2theta)) #
הסבר:
#tan (52.5 ^ @) #
# = sin (52.5 ^ @) / cos (52.5 ^ @) #
# = חטא (105/2) ^ @ / cos (105/2) ^ @ #
# = (2 cdot sin (105/2) ^ @ cdot cos (105/2) ^ @) / (2 cos cos (105/2) ^ @ cdot cos (105/2) ^ @ #
# = חטא (105/2 xx2) ^ @ / (2 cdot cos ^ 2 (105/2) ^ @ #
# = חטא (105) ^ @ / (cos (105) ^ @ 1 +) #
# = sin (60 ^ @ + 45 ^ @) / (cos (60 ^ @ + 45 ^ @) + 1) #
# cdot @ cdot cos45 ^ @ -sin60 ^ @ cdot sin45 ^ @ + 1 # cdot ^ @ cdot cos45 ^ @ cos60 ^ @ cdot sin45 ^ @)
# (sqrt3 / 2 cdot 1 / sqrt2 + 1/2 cdot 1 / sqrt2) / (1/2 cdot 1 / sqrt2-sqrt3 / 2 cdot 1 / sqrt2 + 1 #
# (= sqrt3 + 1) / (2sqrt2)) / ((1-sqrt3 + 2sqrt2) / (2sqrt2) # #
# = (sqrt3 + 1) / (1-sqrt3 + 2sqrt2 #
# 2 (sqrt3 + 1) cdot (1 + 2sqrt2 + sqrt3)) / ((1 + 2sqrt2) ^ 2- (sqrt3) ^ 2) # #
# = (sqrt3 + 2sqrt6 + 3 + 1 + 2sqrt2 + sqrt3) / (1 + 4sqrt2 + 8-3) # #
# = (2 sqrt6 + sqrt3 + sqrt2 + 2)) / (6 + 4sqrt2) # #
# = ((sqrt6 + sqrt3 + sqrt2 + 2)) / (3 + 2sqrt2) # #
# ((3-2sqrt2) (sqrt6 + sqrt3 + sqrt2 + 2)) / ((3 + 2sqrt2) (3-2sqrt2) # #
# = (sqrt6-sqrt3-sqrt2 + 2) / 1 #
# = sqrt6-sqrt3-sqrt2 + 2 #
מקווה שזה עוזר…
תודה…
# tan105 ^ @ = tan (60 ^ @ + 45 ^ @) #
# => tan105 ^ @ = (tan60 ^ @ + tan45 ^ @) / (1-tan60 ^ @ tan45 ^ @) # #
# => tan105 ^ @ = (sqrt3 + 1) / (1-sqrt3 * 1) #
# => tan105 ^ @ = (1 + sqrt3) / (1-sqrt3) # #
# => tan105 ^ @ = - (sqrt3 + 1) (sqrt3-1)) / (sqrt3-1) ^ 2 #
# => tan105 ^ @ = - (3-1) / (4-2sqrt3) #
# => (2tan52.5 ^ @) / (1-tan ^ 2 52.5 ^ @) = = 1 / (2-sqrt3) #
תן #tan52.5^@=x#
עכשיו
# (2x) / (1-x ^ 2) = - 1 / (2-sqrt3) # #
# => x ^ 2-2 (2-sqrt3) x-1 = 0 #
# 2 (2 = 2 = sqrt3) + sqrt (4 (2 sqrt3) ^ 2 + 4)) / 2 #
כפי ש #52.5^@in "ריבוע ראשון-שורש מוזנח" #
# => x = (2 (2-sqrt3) + 2sqrt (2-sqrt3) ^ 2 + 1)) / 2 #
# => x = (2-sqrt3) + sqrt (2-sqrt3) ^ 2 + 1) # #
# => x = (2-sqrt3) + sqrt (8-4sqrt3) #
# => x = (2-sqrt3) + sqrt (2 (4-2sqrt3) #
# => x = (2-sqrt3) + sqrt (2 (sqrt3-1) ^ 2) # #
# => x = 2-sqrt3 + sqrt2 (sqrt3-1) # #
# => x = 2-sqrt3 + sqrt6-sqrt2 #
# => x = sqrt6-sqrt3-sqrt2 + 2 #
