Cum folosiți formulele de reducere a puterii pentru a rescrie expresia sin ^ 8x în ceea ce privește prima putere a cosinusului?

Răspuns:

# Păcat ^ 8x = 1/128 35-56cos2x + 28cos4x-8cos6x + cos8x #

Explicaţie:

# Rarrsin ^ 8x #

# = (2sin ^ 2x) / 2 ^ 4 #

# = 1/16 {1-cos2x} ^ 2 ^ 2 #

# = 1/16 1-2cos2x + cos ^ 2 (2x) ^ 2 #

# = 1/16 (1-2cos2x) ^ 2 + 2 * (1-2cos2x) * cos ^ 2 (2x) + (cos ^ 2 (2x)) ^ 2 #

# = 1/16 1-4cos2x + 4cos ^ 2 (2x) + 2cos ^ 2 (2x) -4cos ^ 3 (2x) + ((2cos ^ 2 (2x)) / 2) ^ 2 #

# = 1/16 1-4cos2x + 6cos ^ 2 (2x) - (3cos (2x) + cos6x) + ((1 + cos4x) / 2) ^ 2 #

# = 1/16 1-4cos2x + 3 * {1 + cos4x} - (3cos (2x) + cos6x) + ((1 + 2cos4x + cos ^ 2 (4x)) / 4) #

# = 1/16 1-4cos2x + 3 + 3cos4x-3cos (2x) -cos6x + ((2 + 4cos4x + 2cos ^ 2 (4x)) / 8) #

# = 1/16 4-7cos2x + 3cos4x-cos6x + ((2 + 4cos4x + 1 + cos8x) / 8) #

# = 1/16 (4-7cos2x + 3cos4x-cos6x + ((3 + 4cos4x + cos8x) / 8) #

# = 1/16 (8 (4-7cos2x + 3cos4x-cos6x) + 3 + 4cos4x + cos8x) / 8 #

# = 1/16 (32-56cos2x + 24cos4x-8cos6x + 3 + 4cos4x + cos8x) / 8 #

# = 1/128 35-56cos2x + 28cos4x-8cos6x + cos8x #