What is (cos(x)+isin(x))^3 ?

The solution to (cos(x)+isin(x))^3 is (4cos^3(x)-3cos(x))+(3sin(x)-4sin^3(x))i

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simplify (cos(x)+isin(x))^3

simplify

(cos (x) + i sin (x))^{3}

(4 cos^{3} (x) - 3 cos (x)) + (3 sin (x) - 4 sin^{3} (x)) i

Solution steps

(cos (x) + i sin (x))^{3}

Apply Perfect Cube Formula:

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

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

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

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

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

(i sin (x))^{3} = - sin^{3} (x) i

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

Rewrite in standard complex form:

(4 cos^{3} (x) - 3 cos (x)) + (3 sin (x) - 4 sin^{3} (x)) i

= (4 cos^{3} (x) - 3 cos (x)) + (3 sin (x) - 4 sin^{3} (x)) i

(- 5, 8) y (- 5, - 6)

(csc (x) + 1) tan (x) - sec (x)

(1 + i)^{9}

(x - 2) (x + 2)

(1 + sin (θ))^{2}

What is (cos(x)+isin(x))^3 ?
The solution to (cos(x)+isin(x))^3 is (4cos^3(x)-3cos(x))+(3sin(x)-4sin^3(x))i