Svar:
# Rarrx = NPI + (- 1) ^ n * (sin ^ (- 1) (3 / sqrt29)) - sin ^ (- 1) (2 / sqrt29) # #n inZZ #
Forklaring:
# Rarr5sinx + 2cosx = 3 #
#rarr (5sinx + 2cosx) / (sqrt (5 ^ 2 + 2 ^ 2)) = 3 / (sqrt (5 ^ 2 + 2 ^ 2) #
# Rarrsinx * (5 / sqrt (29)) + cosx * (2 / sqrt (29)) = 3 / sqrt29 #
Lade # Cosalpha = 5 / sqrt29 # derefter # Sinalpha = sqrt (1-cos ^ 2alpha) = sqrt (1- (5 / sqrt29) ^ 2) = 2 / sqrt29 #
Også, # Alpha = cos ^ (- 1) (5 / sqrt29) = sin ^ (- 1) (2 / sqrt29) #
Nu transformeres den givne ligning til
# Rarrsinx * cosalpha + cosx * sinalpha = 3 / sqrt29 #
#rarrsin (x + alfa) = sin (sin ^ (- 1) (3 / sqrt29)) #
# Rarrx + sin ^ (- 1) (2 / sqrt29) = NPI + (- 1) ^ n * (sin ^ (- 1) (3 / sqrt29)) #
# Rarrx = NPI + (- 1) ^ n * (sin ^ (- 1) (3 / sqrt29)) - sin ^ (- 1) (2 / sqrt29) # #n inZZ #
Svar:
#x = 12 ^ @ 12 + k360 ^ @ #
#x = 124 ^ @ 28 + k360 ^ @ #
Forklaring:
5sin x + 2cos x = 3.
Opdel begge sider med 5.
#sin x + 2/5 cos x = 3/5 = 0,6 # (1)
Opkald #tan t = sin t / (cos t) = 2/5 # --> #t = 21 ^ @ 80 # -> cos t = 0,93.
Ligningen (1) bliver:
#sin x.cos t + sin t.cos x = 0.6 (0.93) #
#sin (x + t) = synd (x + 21,80) = 0,56 #
Kalkulator og enhedscirkel giver 2 løsninger til (x + t) ->
en. x + 21,80 = 33,92
#x = 33,92 - 21,80 = 12 ^ @ 12 #
b. x + 21,80 = 180 - 33,92 = 146,08
#x = 146,08 - 21,80 = 124 ^ @ 28 #
Generelle svar:
#x = 12 ^ @ 12 + k360 ^ @ #
#x = 124 ^ @ 28 + k360 ^ @ #
Tjek efter regnemaskine.
#x = 12 ^ @ 12 # -> 5sin x = 1,05 -> 2cos x = 1,95
5sin x + 2cos x = 1,05 + 1,95 = 3. Bevist.
#x = 124 ^ @ 28 # -> 5sin x = 4,13 -> 2cos x = -1,13
5sin x + 2cos x = 4,13 - 1,13 = 3. Bevist.
