What is (cos(3θ)+isin(3θ))^3 ?

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

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simplify (cos(3θ)+isin(3θ))^3

simplify

(cos (3 θ) + i sin (3 θ))^{3}

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

Solution steps

(cos (3 θ) + i sin (3 θ))^{3}

Apply Perfect Cube Formula:

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

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

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

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

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

(i sin (3 θ))^{3} = - sin^{3} (3 θ) i

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

Rewrite in standard complex form:

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

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

(2 + i)^{18}

- 3 (5 - 4 csc (θ) = csc (θ))

v a l u e (g - f) (3)

lo g_{10} (10000 t)

\frac{( x - 1 ) ^{2} ( y - 3 )}{( 3 x - y ) ^{3}}

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