Patunayan na ang tan20 + tan80 + tan140 = 3sqrt3?

Sagot:

Mangyaring tingnan sa ibaba.

Paliwanag:

Namin, # LHS = tan 20 ^ circ + tan80 ^ circ + tan140 ^ circ #

#color (white) (LHS) = tan20 ^ circ + tan (60 ^ circ + 20 ^ circ) tan (120 ^ circ + 20 ^ circ)

#color (white) (LHS) #=# tan20 ^ circ + (tan60 ^ circ + tan20 ^ circ) / (1-tan60 ^ circtan20 ^ circ) / (1-tan120 ^ circtan20 ^ circ)

Subst. #color (asul) (tan60 ^ circ = sqrt3, tan120 ^ circ = -sqrt3 at tan20 ^ circ = t #

# LHS = t + (sqrt3 + t) / (1-sqrt3t) + (- sqrt3 + t) / (1 + sqrt3t) #

#color (white) (LHS) = t + {(sqrt3 + t) (1 + sqrt3t) + (- sqrt3 + t) (1-sqrt3t)) / ((1-sqrt3t) (1 + sqrt3t)

#color (white) (LHS) = t + (sqrt3 + 3t + t sqrt3t ^ 2-sqrt3 + 3t + t-sqrt3t ^ 2) / (1-3t ^ 2) #

#color (white) (LHS) = t + (8t) / (1-3t ^ 2) #

#color (white) (LHS) = (t-3t ^ 3 + 8t) / (1-3t ^ 2) #

#color (white) (LHS) = (9t-3t ^ 3) / (1-3t ^ 2) #

#color (white) (LHS) = 3 3t-t ^ 3 / (1-3t ^ 2) towhere, kulay (asul) (t = tan20 ^ circ #

#color (white) (LHS) = 3 (3tan20 ^ circ-tan ^ 3 20 ^ circ) / (1-3tan ^ 2 20 ^ circ) #

#color (white) (LHS) = 3 tan3 (20 ^ circ) toApply (2) # para sa # theta = 20 ^ circ #

# LHS = 3tan60 ^ circ #

# LHS = 3sqrt3 = RHS #

Tandaan:

# (1) kayumanggi (A + B) = (tanA + tanB) / (1-tanAtanB) #

# (2) tan3theta = (3tantheta-tan ^ 3theta) / (1-3tan ^ 2theta) #

# LHS = tan20 + tan80 + tan140 #

# = tan20 + tan80 + tan (180-40) #

# = tan20 + tan80-tan 40 #

# = tan20 + sin 80 / cos 80-sin 40 / cos 40 #

# = sin 20 / cos 20+ (sin 80cos 40-cos 80sin 40) / (cos 80cos 40) #

# = (sin 20cos 80cos 40 + sin 40cos 20) / (cos 20cos 80cos 40) #

Ngayon denominador ng pananalitang ito

# = cos 20cos 80cos 40 #

# = (4 * 2sin 20cos 20cos 40cos 80) / (8sin 20) #

# = (2 * 2sin 40cos 40cos 80) / (8sin 20) #

# = (2sin 80cos 80) / (8sin 20) #

# = (kasalanan 160) / (8sin 20) #

# = (kasalanan (180-20)) / (8sin 20) #

# = (kasalanan 20) / (8sin 20) #

#=1/8#

Kaya nga

# LHS = 8 (kasalanan 20cos 80cos 40 + kasalanan 40cos 20) #

# = 4sin 20 * (2cos 80cos 40) + 4 * 2sin 40cos 20 #

# = 4sin 20 (cos 120 + cos 40) +4 (kasalanan 60 + kasalanan 20) #

# = 4sin 20 (-1 / 2 + cos 40) +4 (sqrt3 / 2 + kasalanan 20) #

# = - 2sin 20 + 4sin 20cos 40 + 2sqrt3 + 4sin 20 #

# = 4sin 20cos 40 + 2sqrt3 + 2sin 20 #

# = 2 (kasalanan 60-kasalanan 20) + 2sqrt3 + 2sin 20 #

# = 2 (sqrt3 / 2-sin 20) + 2sqrt3 + 2sin 20 #

# = sqrt3-2sin 20 + 2sqrt3 + 2sin 20 #

# = 3sqrt3 #

Isang nakakatawa na diskarte na ginagamit ang anwer # 3sqrt3 # ibinigay.

Maaari naming isulat ang LHS bilang mga sumusunod na alam namin # sqrt3 = tan 60 #

# LHS = tan 20 + tan 80 + tan 140 #

# = 3sqrt3 + (tan 20-tan 60) + (tan 80-tan 60) + (tan 140-tan 60) #

# = 3sqrt3 + (tan 20-tan 60) + (tan 80-tan 60) + (tan (180-40) -tan 60) #

# = 3sqrt3 + (tan 20-tan 60) + (tan 80-tan 60) - (kayumanggi 40 + tan 60) #

(Sin 40 / cos40 + sin 60 / cos60) #

# = 3sqrt3-sin (60-20) / (cos 20cos60) + kasalanan (80-60) / (cos 80cos60) -sin (60 + 40) / (cos40cos60) #

# = 3sqrt3- (2sin 40) / cos 20+ (2sin 20) / cos 80- (2sin 100) / cos 40 #

# = 3sqrt3- (4sin 20cos 20) / cos 20+ (4sin 10 cos 10) / sin 10- (4sin 40cos 40) / cos 40 #

# = 3sqrt3-4sin 20 + 4cos 10-4sin 40 #

# = 3sqrt3-4 (kasalanan 20 + kasalanan 40) + 4cos 10 #

# = 3sqrt3-4 (2 kasalanan 30cos1 0) + 4cos 10 #

# = 3sqrt3-4 (2 * 1/2 * cos1 0) + 4cos 10 #

# = 3sqrt3-4cos 10 + 4cos 10 #

# = 3sqrt3 #

Sagot:

Paliwanag sa ibaba

Paliwanag:

# x = tan20 + tan80 + tan140 #

=# sin20 / cos20 + sin80 / cos80 + tan (180-40) #

=# (cos80 * sin20 + sin80 * cos20) / (cos80 * cos20) -tan40 #

=#sin (80 + 20) / (cos80 * cos20) -sin40 / cos40 #

=# sin100 / (cos80 * cos20) -sin40 / cos40 #

=# sin80 / (cos80 * cos20) -sin40 / cos40 #

=# (sin80 * cos40-cos80 * sin40 * cos20) / (cos80 * cos40 * cos20) #

=# (sin20 * (8sin80 * cos40-8cos80 * sin40 * cos20)) / (8cos80 * cos40 * cos20 * sin20) #

=# (sin20 * (4sin120 + 4sin40-4cos20 * (sin120-sin40))) / (4cos80 * cos40 * sin40) #

=# (sin20 * (4sin120 + 4sin40-4sin120 * cos20 + 4sin40 * cos20)) / (2cos80 * sin80) #

=# (sin20 * (4sin60 + 4sin40-4sin60 * cos20 + 4sin40 * cos20)) / (sin160) #

=# (sin20 * (4sin60 + 4sin40-2sin80-2sin40 + 2sin60 + 2sin20)) / (sin20) #

=# 6sin60 + 2sin40-2sin80 + 2sin20 #

=# 3sqrt3 + 2sin20- (2sin80-2sin40) #

=# 3sqrt3 + 2sin20-4cos60 * sin20 #

=# 3sqrt3 + 2sin20-2sin20 #

=# 3sqrt3 #