What is (cos(θ)+sin(θ)i)^3 ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (cos(θ)+sin(θ)i)^3

simplify

(cos (θ) + sin (θ) i)^{3}

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

Solution steps

(cos (θ) + sin (θ) i)^{3}

Apply Perfect Cube Formula:

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

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

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

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

(sin (θ) i)^{3} = - 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:

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

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

4 (6 + 10 + 14)

\frac{( x - 2 ) ( 12 - x )}{x - 11}

(x^{2} + 3 x + 2)^{5}

2 (1 + sin (x)) (1 + cos (x))

(z - r cos (θ))^{2}

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