How do you divide #( i+3) / (-i +9 )# in trigonometric form?

1 Answer

In trigonometric form: #0.35(cos 5.85 +i sin 5.85)#

Explanation:

# (3+i)/(9-i)# #Z=a+ib #. Modulus: #|Z|=sqrt (a^2+b^2)#;

Argument: #theta=tan^-1(b/a)# Trigonometrical form :

#Z =|Z|(costheta+isintheta);Z_1= 3+ i #.

Modulus:#|Z_1|=sqrt(3^2+1^2)~~ 3.16 #

Argument: #tan alpha= (|1|)/(|3|):. alpha = tan^-1(1/3)~~0.32#

#Z_1# lies on first quadrant, so #theta =alpha ~~ 0.32#

# :. Z_1=3.16(cos 0.32+isin 0.32) #

#Z_2= 9 - i #. Modulus:#|Z_2|=sqrt(9^2+1^2) #

#=sqrt 82~~ 9.06# Argument: #tan alpha= (|-1|)/(|9|)#

#=1/9 :.alpha =tan^-1 (1/9) = 0.11 ; Z_2# lies on fourth

quadrant.#:. theta=2pi-alpha ~~6.17#

# :. Z_2=9.06(cos 6.17+isin 6.17) :. (3+i)/(9-i) =#

# Z= (3.16(cos0.32+isin 0.32))/(9.06(cos 6.17+isin6.17)#

#Z=0.35(cos(0.32-6.17)+isin (0.32-6.17))# or

#Z=0.35(cos 5.85 +i sin 5.85) =13/41+6/41 i#

In trigonometric form; #0.35(cos 5.85 +i sin 5.85)#