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# Trigonometry Problem

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599
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+1884

Find all of the fifth roots of the complex number $$4+32i$$.  Put your answers in the rectangular form $$a+bi$$ and in the polar form $$z=re^(i\Theta )$$.  Please show how you got to your answers.

Nov 24, 2015

#2
+1884
+10

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

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

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

$$r=\sqrt(16+1024)$$

$$r=\sqrt1040$$

$$r=4\sqrt65$$

$$tan(\Theta)=b/a$$

$$tan(\Theta)=32/4$$

$$tan(\Theta)=8$$

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

$$\Theta ≈1.4464413322481$$

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

$$z≈4\sqrt65*e^(i*1.4464413322481)$$

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

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

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

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

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

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

$$z≈0.9500952169545+i*1.763445966914$$

$$z≈0.9500952169545+1.763445966914i$$

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

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

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

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

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

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

$$z≈-0.049814942634829+i*0.0001545372385216$$

$$z≈-0.049814942634829+0.0001545372385216i$$

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

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

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

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

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

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

$$z≈-0.24920377983349+i*1.9875412136019$$

$$z≈-0.24920377983349+1.9875412136019i$$

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

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

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

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

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

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

$$z≈-1.2173308220295+i*1.5907634234047$$

$$z≈-1.2173308220295+1.5907634234047i$$

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

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

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

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

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

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

$$z≈-1.1367300079339+i*-1.6493233426371$$

$$z≈-1.1367300079339+-1.6493233426371i$$

$$z≈-1.1367300079339-1.6493233426371i$$

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Nov 26, 2015

#1
+5

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

Nov 24, 2015
#2
+1884
+10

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

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

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

$$r=\sqrt(16+1024)$$

$$r=\sqrt1040$$

$$r=4\sqrt65$$

$$tan(\Theta)=b/a$$

$$tan(\Theta)=32/4$$

$$tan(\Theta)=8$$

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

$$\Theta ≈1.4464413322481$$

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

$$z≈4\sqrt65*e^(i*1.4464413322481)$$

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

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

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

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

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

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

$$z≈0.9500952169545+i*1.763445966914$$

$$z≈0.9500952169545+1.763445966914i$$

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

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

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

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

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

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

$$z≈-0.049814942634829+i*0.0001545372385216$$

$$z≈-0.049814942634829+0.0001545372385216i$$

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

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

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

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

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

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

$$z≈-0.24920377983349+i*1.9875412136019$$

$$z≈-0.24920377983349+1.9875412136019i$$

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

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

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

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

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

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

$$z≈-1.2173308220295+i*1.5907634234047$$

$$z≈-1.2173308220295+1.5907634234047i$$

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

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

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

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

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

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

$$z≈-1.1367300079339+i*-1.6493233426371$$

$$z≈-1.1367300079339+-1.6493233426371i$$

$$z≈-1.1367300079339-1.6493233426371i$$

gibsonj338 Nov 26, 2015