Trigonometry Problem

$(4+32i)^(1/5)$

 

$r=\sqrt(a^2+b^2)$

 

$r=\sqrt(4^2+32^2)$

 

$r=\sqrt(16+1024)$

 

$r=\sqrt1040$

 

$r=4\sqrt65$

 

$tan(\Theta)=b/a$

 

$tan(\Theta)=32/4$

 

$tan(\Theta)=8$

 

$\Theta=tan^-1(8)$

 

$\Theta ≈1.4464413322481$

 

 

$z=r*e^(i*\Theta)$

 

$z≈4\sqrt65*e^(i*1.4464413322481)$

 

$z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)$

 

$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))$

 

$z^(1/5)≈2.003103242348*e^(i*1.0766143512748)$

 

$z=r*(cos(\Theta)+i*sin(\Theta))$

 

$z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))$

 

$z≈2.003103242348*(0.4743116564868+i*0.88035700289064)$

 

$z≈0.9500952169545+i*1.763445966914$

 

$z≈0.9500952169545+1.763445966914i$

 

 

$z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)$

 

$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))$

 

$z^(1/5)≈2.003103242348*e^(i*1.5459253278855)$

 

$z=r*(cos(\Theta)+i*sin(\Theta))$

 

$z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))$

 

$z≈2.003103242348*(0.024868434927217+i*0.99969073264899)$

 

$z≈-0.049814942634829+i*0.0001545372385216$

 

$z≈-0.049814942634829+0.0001545372385216i$

 

 

$z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)$

 

$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))$

 

$z^(1/5)≈2.003103242348*e^(i*1.6955283616309)$

 

$z=r*(cos(\Theta)+i*sin(\Theta))$

 

$z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))$

 

$z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)$

 

$z≈-0.24920377983349+i*1.9875412136019$

 

$z≈-0.24920377983349+1.9875412136019i$

 

 

$z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)$

 

$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))$

 

$z^(1/5)≈2.003103242348*e^(i*4.059994507574)$

 

$z=r*(cos(\Theta)+i*sin(\Theta))$

 

$z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))$

 

$z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)$

 

$z≈-1.2173308220295+i*1.5907634234047$

 

$z≈-1.2173308220295+1.5907634234047i$

 

 

$z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)$

 

$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))$

 

$z^(1/5)≈2.003103242348*e^(i*5.315836512196)$

 

$z=r*(cos(\Theta)+i*sin(\Theta))$

 

$z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))$

 

$z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)$

 

$z≈-1.1367300079339+i*-1.6493233426371$

 

$z≈-1.1367300079339+-1.6493233426371i$

 

$z≈-1.1367300079339-1.6493233426371i$

.

z = (4 + 32i)^(1/5)  

Divide: 1 / 5 =0.2

Power: (4+32i) ^ 0.2 = 1.9198686+0.5714255i

Algebraic form:
z = 1.9198686+0.5714255i

Exponential form:
z = 2.0031032 × ei 16°34'30″

Trigonometric form:
z = 2.0031032 × (cos 16°34'30″ + i sin 16°34'30″)

Polar form:
r = |z| = 2.0031
φ = arg z = 16.575° = 16°34'30″ = 0.09208π