Antwoord:
Zie onder.
Uitleg:
Wij nemen, # LHS = tan 20 ^ circ + tan80 ^ circ + tan140 ^ circ #
#color (wit) (LHS) = tan20 ^ cir + tg (60 ^ ci + 20 ^ circ) + tan (120 ^ ci + 20 ^ circ) #
#color (wit) (LHS) #=# Tan20 ^ cir + (tan60 ^ ci + tan20 circ ^) / (1-tan60 circtan20 ^ ^ circ) + (tan120 ^ ci + tan20 circ ^) / (1-tan120 ^ ^ circtan20 circ) #
Subst. #color (blauw) (tan60 ^ circ = sqrt3, tan120 ^ circ = -sqrt3 en tan20 ^ circ = t #
# LHS = t + (sqrt3 + t) / (1-sqrt3t) + (- sqrt3 + t) / (1 + sqrt3t) #
#color (wit) (LHS) = t + {(sqrt3 + t) (1 + sqrt3t) + (- sqrt3 + t) (1-sqrt3t)) / ((1-sqrt3t) (1 + sqrt3t)) #
#color (wit) (LHS) = t + (sqrt3 + 3t + t + sqrt3t ^ 2-sqrt3 + 3t + t-sqrt3t ^ 2) / (^ 1-3T 2) #
#color (wit) (LHS) = t + (8t) / (^ 1-3T 2) #
#color (wit) (LHS) = (t-3t ^ 3 + 8t) / (^ 1-3T 2) #
#color (wit) (LHS) = (9t-3t ^ 3) / (^ 1-3T 2) #
#color (wit) (LHS) = 3 (3t-t ^ 3) / (^ 1-3T 2) towhere, kleur (blauw) (t = tan20 ^ circ #
#color (wit) (LHS) = 3 (3tan20 ^ circ-tan ^ 3 20 ^ circ) / (1-3tan ^ 2 20 ^ circ) #
#color (wit) (LHS) = 3 tan3 (20 ^ circ) toapply (2) # voor # Theta = 20 ^ circ #
# LHS = 3tan60 ^ circ #
# LHS = 3sqrt3 = RHS #
Notitie:
# (1) tan (A + B) = (tanA + tanB) / (1-tanAtanB) #
# (2) tan3theta = (3tantheta-tan ^ 3theta) / (1-3tan ^ 2 theta) #
# 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) #
Nu noemer van deze uitdrukking
# = cos 20cos 80cos 40 #
# = (4 * 2sin 20cos 20cos 40cos 80) / (8sin 20) #
# = (2 * 2sin 40cos 40cos 80) / (8sin 20) #
# = (2sin 80cos 80) / (8sin 20) #
# = (sin 160) / (8sin 20) #
# = (sin (180-20)) / (8sin 20) #
# = (sin 20) / (8sin 20) #
#=1/8#
Vandaar
# LHS = 8 (sin 20cos 80cos 40 + sin 40cos 20) #
# = 4sin 20 * (2cos 80cos 40) + 4 * 2sin 40cos 20 #
# = 4sin 20 (cos 120 + cos 40) +4 (sin 60 + sin 20) #
# = 4sin 20 (-1 / 2 + cos 40) +4 (sqrt3 / 2 + sin 20) #
# = - 2sin 20 + 4sin 20cos 40 + 2sqrt3 + 4sin 20 #
# = 4sin 20cos 40 + 2sqrt3 + 2sin 20 #
# = 2 (sin 60-sin 20) + 2sqrt3 + 2sin 20 #
# = 2 (sqrt3 / 2-sin 20) + 2sqrt3 + 2sin 20 #
# = sqrt3-2sin 20 + 2sqrt3 + 2sin 20 #
# = 3sqrt3 #
Een grappige benadering met behulp van de anwer # 3sqrt3 # gegeven.
We kunnen LHS als volgt schrijven zoals we dat weten # 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) - (tan 40 + tan 60) #
# = 3sqrt3 + (sin 20 / cos 20-sin 60 / cos60) + (sin 80 / cos 80-sin 60 / cos60) - (sin 40 / cos40 + sin 60 / cos60) #
# = 3sqrt3-sin (60- 20) / (cos 20cos60) + sin (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 (sin 20 + sin 40) + 4cos 10 #
# = 3sqrt3-4 (2 sin 30cos1 0) + 4cos 10 #
# = 3sqrt3-4 (2 * 1/2 * cos1 0) + 4cos 10 #
# = 3sqrt3-4cos 10 + 4cos 10 #
# = 3sqrt3 #
Antwoord:
Uitleg hieronder
Uitleg:
# 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 #
