Finden Sie den Wert von sin (a + b), wenn tan a = 4/3 und cot b = 5/12, 0 ° Grad

Antworten:

#sin (a + b) = 56/65 #

Gegeben, # tana = 4/3 und cotb = 5/12 #

# rarrcota = 3/4 #

# rarrsina = 1 / csca = 1 / sqrt (1 + cot ^ 2a) = 1 / sqrt (1+ (3/4) ^ 2) = 4/5 #

# rarrcosa = sqrt (1-sin ^ 2a) = sqrt (1- (4/5) ^ 2) = 3/5 #

# rarrcotb = 5/12 #

# rarrsinb = 1 / cscb = 1 / sqrt (1 + cot ^ 2b) = 1 / sqrt (1+ (5/12) ^ 2) = 12/13 #

# rarrcosb = sqrt (1-sin ^ 2b) = sqrt (1- (12/13) ^ 2) = 5/13 #

Jetzt, #sin (a + b) = sina * cosb + cosa * sinb #

#=(4/5)(5/13)+(3/5)*(12/13)=56/65#

#sin (a + b) = 56/65 #

Hier, # 0 ^ circ <color (violett) (a) <90 ^ circ => I ^ (st) Quadrant => color (blau) (Alle, fns.> 0. #

# 0 ^ circ <color (violett) (b) <90 ^ circ => I ^ (st) Quadrant => color (blau) (All, fns.> 0 #

So, # 0 ^ circ <Farbe (violett) (a + b) <180 ^ circ => I ^ (st) und II ^ (nd) Quadrant #

# => Farbe (blau) (sin (a + b)> 0 #

Jetzt, # tana = 4/3 => seca = + sqrt (1 + tan ^ 2a) = sqrt (1 + 16/9) = 5/3 #

#:. Farbe (Rot) (Cosa) = 1 / Sek. = Farbe (Rot) (3/5 #

# => Farbe (rot) (sina) = + sqrt (1-cos ^ 2a) = sqrt (1-9 / 25) = Farbe (rot) (4/5 #

Ebenfalls, # cotb = 5/12 => cscb = + sqrt (1 + cot ^ 2b) = sqrt (1 + 25/144) = 13/12 #

#:. farbe (rot) (sinb) = 1 / cscb = farbe (rot) (12/13 #

# => Farbe (rot) (cosb) = + sqrt (1-sin ^ 2b) = sqrt (1-144 / 169) = Farbe (rot) (5/13 #

Daher, #sin (a + b) = sinacosb + cosasinb #

# => sin (a + b) = 4/5xx5 / 13 + 3 / 5xx12 / 13 #

#sin (a + b) = 20/65 + 36/65 = 56/65 #