Question

Выполните действие в тригонометрической форме

Answer 1

(i² -i√3)³ / (1-i)²⁶) = (-1 -i√3)³ / (1 -i)²⁶  =( -(1+i √3))³ /( 1 - i)²⁶ = -(1+i√3)/(1 -i)²⁶ =
-(2(cosπ/3 +isinπ/3))³/√(2(cosπ/4 -isinπ/4))²⁶ = 
-2³(cos3*π/3 + isin3*π/3) /2¹³(cos26*π/4 -isin26*π/4) = 
-8(cosπ + isinπ) /2¹³(cos13π/2 -isin13π/2)  = -8(-1+0)/2¹³(0 -i) =-2³/2¹³i = (1/21⁰)i.
* * * * * *
z =a+ib ; z =r(cosα +  i sinα )    ;     r =√(a²+b²) ;  α =arctq(b/a)
(r(cosα+isinα) ) ^n  =r^k(cosnα +i sinnα) ; 
(r₁(cosα₁+isinα₁)*r₂(cosα₂+isinα₂) =(r₁*r₂) (cos(α₁+α₂) +isin(α₁+α₂)) ;
(r₁(cosα₁+isinα₁)/r₂(cosα₂+isinα₂) =(r₁/r₂) (cos(α₁-α₂) +isin(α₁-α₂)) ;
***************************
z₁ =(1+i√3) ,
модуль этого  числа:  r₁ =√(1² +(√3)²) =√(1 +3)=2;
аргумент  этого  числа : tqα =b/a =√3/1=√3 ⇒α=60° или α= π/3 радиан.
z₁ =(1+i√3)  =2(cosπ/3 +isinπ/3) .

Answer 2

Trigonometric Form

To express a complex number in trigonometric form, we can use the polar form, which represents the number as a magnitude and an angle. The polar form of a complex number is given by:

z = r(cosθ + isinθ)

where: - z is the complex number - r is the magnitude or modulus of the complex number - θ is the argument or angle of the complex number

To convert a complex number from rectangular form (a + bi) to trigonometric form, we can use the following formulas:

r = √(a^2 + b^2)

θ = arctan(b/a)

Let's take an example to illustrate this.

Example: Convert the complex number 3 + 4i to trigonometric form.

Solution: Using the formulas mentioned above, we can calculate the magnitude and angle as follows:

r = √(3^2 + 4^2) = √(9 + 16) = √25 = 5

θ = arctan(4/3) ≈ 53.13°

Therefore, the complex number 3 + 4i can be expressed in trigonometric form as 5(cos(53.13°) + isin(53.13°)).

Please note that the angle θ is usually given in radians, but here we have provided it in degrees for simplicity.