Bewijs dat cot (A / 2) - 3cot ((3A) / 2) = (4sinA) / (1 + 2cosA)?

Antwoord:

Raadpleeg de Uitleg.

Uitleg:

We weten dat, # Tan3theta = (3tantheta-tan ^ 3theta) / (1-3tan ^ 2 theta) #.

#:. cot3theta = 1 / (tan3theta) = (1-3tan ^ 2 theta) / (3tantheta-tan ^ 3theta) #

#:. kinderbed ((3A) / 2) = {1-3tan ^ 2 (A / 2)} / {3tan (A / 2) -tan ^ 3 (A / 2)} #.

Letting #tan (A / 2) = t, # wij hebben,

#cot (A / 2) -3cot ((3A) / 2) #, # = 1 / t-3 {(1-3T ^ 2) / (3t-t ^ 3)} #, 1 # / T {3 (1-3T ^ 2)} / {t (3-t ^ 2)} #, # = {(3-t ^ 2) -3 (1-3T ^ 2)} / {t (3-t ^ 2)} #, # = (8t ^ annuleren (2)) / {annuleren (t) (3-t ^ 2)} #, # = (8t) / {(1 + t ^ 2) 2 (1-t ^ 2)} #

# = {4 * (2t) / (1 + t ^ 2)} / {(1 + t ^ 2) / (1 + t ^ 2) + 2 * (1-t ^ 2) / (1 + t ^ 2)} #.

Let daar op, # (2t) / (1 + t ^ 2) = {2tan (A / 2)} / (1 + tan ^ 2 (A / 2)) = sinA, en #

# (1-t ^ 2) / (1 + t ^ 2) = cosa #.

#rArrcot (A / 2) -3cot ((3A) / 2) = (4sinA) / (1 + 2cosA), "zoals gewenst!" #

Antwoord:

Zie onder.

Uitleg:

# LHS kinderbed = (x / 2) -3cot ((3x) / 2) #

# = Cos (x / 2) / sin (x / 2) -3 * cos ((3x) / 2) / sin ((3x) / 2) #

# = (Sin ((3x) / 2) * cos (x / 2) -3 * cos ((3x) / 2) * sin (x / 2)) / (sin (x / 2) * sin ((3x) / 2) #

# = (2sin ((3x) / 2) * cos (x / 2) * -3 2cos ((3x) / 2) * sin (x / 2)) / (2sin (x / 2) * sin ((3x) / 2) #

# = (Sin ((3x) / 2 + x / 2) + sin ((3x) / 2-x / 2) * -3 {sin ((3x) / 2 + x / 2) sin ((3x) / 2-x / 2)}) / (cos ((3x) / 2-x / 2) -cos ((3x) / 2 + x / 2) #

# = (Sin ((4x) / 2) + sin ((2x) / 2) * -3 {sin ((4x) / 2) sin ((2x) / 2)}) / (cos ((2x) / 2) -cos ((4x) / 2) #

# = (Sin2x + SiNx-3sin2x 3sinx +) / (cosx-cos2x) #

# = (4sinx-2sin2x) / (cosx- (cos ^ 2x-^ sin 2x)) #

# = (4sinx-4sinx * cosx) / (cosx-cos ^ 2x + sin ^ 2x) #

# = (4sinx (1-cosx)) / (cosx (1-cosx) + (1-cosx) (1 + cosx)) #

# = (4sinx (1-cosx)) / ((1-cosx) (cosx + 1 + cosx) #

# = (4sinx) / (1 + 2cosx) = RHS #

# LHS = kinderbed (A / 2) -3cot ((3A) / 2) #

# = Cos (A / 2) / sin (A / 2) -cos ((3A) / 2) / sin ((3A) / 2) -2cot ((3A) / 2) #

# = (Sin ((3A) / 2) * cos (A / 2) -cos ((3A) / 2) * sin (A / 2)) / (sin (A / 2) * sin ((3A) / 2)) - 2cot ((3A) / 2) #

# = sin (A) / (sin (A / 2) * sin ((3A) / 2)) -2cot ((3A) / 2) #

# = (2sin (A / 2) cos (A / 2)) / (sin (A / 2) * sin ((3A) / 2)) -2cot ((3A) / 2) #

# = 2cos (A / 2) / sin ((3A) / 2) -2 * cos ((3A) / 2) / sin ((3 A) / 2) #

# = 2 (cos (A / 2) -cos ((3A) / 2)) / sin ((3 A) / 2) #

# = 2 (2sin (A / 2) sin (A)) / (3sin (A / 2) -4sin ^ 3 (A / 2)) #

# = (4sin (A / 2) sin (A)) / (sin (A / 2) (3-4sin ^ 2 (A / 2)) #

# = (4sin (A)) / (3-2 (1-cosA)) #

# = (4sin (A)) / (1 + 2cosA) = RHS #