SOLUTION: Let tan(A)=1/3 where {{{0<A<pi/2}}} a) Show that 4A = {{{tan^-1(24/7)}}} b) Given that {{{0<A<pi/2}}}, express z = 7 + 24i in the form {{{rcis(theta)}}} giving all the possible v

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson  -> Lesson -> SOLUTION: Let tan(A)=1/3 where {{{0<A<pi/2}}} a) Show that 4A = {{{tan^-1(24/7)}}} b) Given that {{{0<A<pi/2}}}, express z = 7 + 24i in the form {{{rcis(theta)}}} giving all the possible v      Log On


   

Question 1152857: Let tan(A)=1/3 where 0%3CA%3Cpi%2F2
a) Show that 4A = tan%5E-1%2824%2F7%29
b) Given that 0%3CA%3Cpi%2F2, express z = 7 + 24i in the form rcis%28theta%29 giving all the possible values of theta in terms of A.
c) Hence obtain in the form a + ib, the four fourth roots of z.
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will answer part  (a),  ONLY. 


Use the formula for tan(2A)


    tan(2A) = %28tan%28A%29%2Btan%28A%29%29%2F%281-tan%5E2%28A%29%29 = %282%2Atan%28A%29%29%2F%281-tan%5E2%28A%29%29 = %28%282%2F3%29%29%2F%28%281-%281%2F3%29%5E2%29%29 = %28%282%2F3%29%29%2F%28%288%2F9%29%29 = 3%2F4.


Use the formula for tan(2A), again


    tan(4A) = %28tan%282A%29%2Btan%282A%29%29%2F%281-tan%5E2%282A%29%29 = %282%2Atan%282A%29%29%2F%281-tan%5E2%282A%29%29 = %28%286%2F4%29%29%2F%28%281-%283%2F4%29%5E2%29%29 = %28%286%2F4%29%29%2F%28%287%2F16%29%29 = 24%2F7.


Therefore,  4A = arctan%2824%2F7%29.

Solved.



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