Cos ¹ (sqrtcos α) tan ¹ (sqrtcos α) = x, potom aká je hodnota sin x?

odpoveď:

# Sinx = tan (alfa / 2) -cosalpha / (sqrt2cos (alfa / 2)) #

vysvetlenie:

nechať # Sqrtcosalpha = m #

#rarrcos ^ (- 1), (m) -topenia ^ (- 1), (m) = x #

nechať #cos ^ (- 1), m = y # potom # Útulný = m #

# Rarrsiny = sqrt (1-cos ^ 2y) = sqrt (1-m ^ 2) #

# Rarr = sin ^ (- 1) (sqrt (1-m ^ 2)) = cos ^ (- 1) m #

Tiež, nech #tan ^ (- 1), m = z # potom # Tanz = m #

# Rarrsinz = 1 / cscz = 1 / sqrt (1 + postieľka ^ 2z) = 1 / sqrt (1+ (1 / m) ^ 2) = m / sqrt (1 + m ^ 2) #

# Rarrz = sin ^ (- 1) (m / sqrt (1 + m ^ 2)) = tan ^ (- 1) m #

#rarrcos ^ (- 1), (m) -topenia ^ (- 1), (m) #

# = Sin ^ (- 1) (sqrt (1-m ^ 2)) - sin ^ (- 1) (m / sqrt (1 + m ^ 2)) #

# = Sin ^ -1 (sqrt (1-m ^ 2) * sqrt (1- (m / sqrt (1 + m ^ 2)) ^ 2) - (m / sqrt (1 + m ^ 2)) * sqrt (1- (sqrt (1-m ^ 2)) ^ 2)) #

# = Sin ^ (- 1) (sqrt ((1-cosalpha) / (1 + cosalpha)) - cosalpha / sqrt (1 + cosalpha)) #

# = Sin ^ (- 1) (tan (alfa / 2) -cosalpha / (sqrt2cos (alfa / 2))) = x #

# Rarrsinx = sin (sin ^ (- 1) (tan (alfa / 2) -cosalpha / (sqrt2cos (alfa / 2)))) = tan (alfa / 2) / (-cosalpha sqrt2cos (alfa / 2)) #