What is (cos(θ)+isin(θ))^2(1+2cos(θ)) ?

The solution to (cos(θ)+isin(θ))^2(1+2cos(θ)) is (cos(2θ)+2cos(2θ)cos(θ))+(sin(2θ)+2cos(θ)sin(2θ))i

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simplify (cos(θ)+isin(θ))^2(1+2cos(θ))

simplify

(cos (θ) + i sin (θ))^{2} (1 + 2 cos (θ))

(cos (2 θ) + 2 cos (2 θ) cos (θ)) + (sin (2 θ) + 2 cos (θ) sin (2 θ)) i

Solution steps

(cos (θ) + i sin (θ))^{2} (1 + 2 cos (θ))

Expand

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

= (i sin (2 θ) + cos (2 θ)) (1 + 2 cos (θ))

Apply FOIL method

(i sin (2 θ) + cos (2 θ)) (1 + 2 cos (θ)) : sin (2 θ) i + 2 cos (θ) sin (2 θ) i + cos (2 θ) + 2 cos (2 θ) cos (θ)

= sin (2 θ) i + 2 cos (θ) sin (2 θ) i + cos (2 θ) + 2 cos (2 θ) cos (θ)

Rewrite in standard complex form:

(cos (2 θ) + 2 cos (2 θ) cos (θ)) + (sin (2 θ) + 2 cos (θ) sin (2 θ)) i

= (cos (2 θ) + 2 cos (2 θ) cos (θ)) + (sin (2 θ) + 2 cos (θ) sin (2 θ)) i

(3 x - 5 y = 4,) (3 x - 2 y = 12)

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

\frac{1}{s ( s ^{3} + 6 s ^{2} + 11 x + 6 )}

- e^{3 x} (e^{- x} + 1)

(1 + 2 π sin (2 π t))^{2}

What is (cos(θ)+isin(θ))^2(1+2cos(θ)) ?
The solution to (cos(θ)+isin(θ))^2(1+2cos(θ)) is (cos(2θ)+2cos(2θ)cos(θ))+(sin(2θ)+2cos(θ)sin(2θ))i