What is expand (cos(θ)+isin(θ))^3 ?

The solution to expand (cos(θ)+isin(θ))^3 is (cos^3(θ)-3sin^2(θ)cos(θ))+(3cos^2(θ)sin(θ)-sin^3(θ))i

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expand (cos(θ)+isin(θ))^3

expand

(cos (θ) + i sin (θ))^{3}

(cos^{3} (θ) - 3 sin^{2} (θ) cos (θ)) + (3 cos^{2} (θ) sin (θ) - sin^{3} (θ)) i

Solution steps

(cos (θ) + i sin (θ))^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(cos (θ) + i sin (θ))^{3} = cos^{3} (θ) + 3 cos^{2} (θ) i sin (θ) + 3 cos (θ) (i sin (θ))^{2} + (i sin (θ))^{3}

= cos^{3} (θ) + 3 cos^{2} (θ) i sin (θ) + 3 cos (θ) (i sin (θ))^{2} + (i sin (θ))^{3}

Simplify

cos^{3} (θ) + 3 cos^{2} (θ) i sin (θ) + 3 cos (θ) (i sin (θ))^{2} + (i sin (θ))^{3} : cos^{3} (θ) + 3 cos^{2} (θ) sin (θ) i - 3 sin^{2} (θ) cos (θ) - sin^{3} (θ) i

= cos^{3} (θ) + 3 cos^{2} (θ) sin (θ) i - 3 sin^{2} (θ) cos (θ) - sin^{3} (θ) i

Rewrite in standard complex form:

(cos^{3} (θ) - 3 sin^{2} (θ) cos (θ)) + (3 cos^{2} (θ) sin (θ) - sin^{3} (θ)) i

= (cos^{3} (θ) - 3 sin^{2} (θ) cos (θ)) + (3 cos^{2} (θ) sin (θ) - sin^{3} (θ)) i

\frac{\partial}{\partial x} ((f (x, y)) (x^{2} + y^{2}))

sin (x) (1 + tan (x))

(a + b + c + d + e)^{3}

\frac{5 ( 1 + 5 )}{2}

\frac{x ^{3}}{( 7 + 14 x ^{2} ) ^{2}}

What is expand (cos(θ)+isin(θ))^3 ?
The solution to expand (cos(θ)+isin(θ))^3 is (cos^3(θ)-3sin^2(θ)cos(θ))+(3cos^2(θ)sin(θ)-sin^3(θ))i