Answer to Question #212671 in Complex Analysis for Faith

Question #212671

Use de moivres theorem to

1.derive the 4th roots of w=-8i.

2.express cos(4@) and sin(5@) in terms of powers of cos@ and sin@.

3.expand cos^6@ in terms of multiple powers of z based on @.

4.express cos^3@sin^4@ in terms of multiple angles.

NOTE:@ represents theta.




1
Expert's answer
2021-07-02T14:27:10-0400

1.


8i=8(cos(π2)+isin(π2))-8i=8(\cos(-\dfrac{\pi}{2})+i\sin(-\dfrac{\pi}{2}))

k=0:k=0:

84(cos(π2+2π(0)4)+isin(π2+2π(0)4))\sqrt[4]{8}(\cos(\dfrac{-\dfrac{\pi}{2}+2\pi(0)}{4})+i\sin(\dfrac{-\dfrac{\pi}{2}+2\pi(0)}{4}))

=23/4(cos(π8)+isin(π8))=2^{3/4}(\cos(-\dfrac{\pi}{8})+i\sin(-\dfrac{\pi}{8}))

=21/42+2i21/422=2^{-1/4}\sqrt{2+\sqrt{2}}-i2^{-1/4}\sqrt{2-\sqrt{2}}

k=1:k=1:

84(cos(π2+2π(1)4)+isin(π2+2π(1)4))\sqrt[4]{8}(\cos(\dfrac{-\dfrac{\pi}{2}+2\pi(1)}{4})+i\sin(\dfrac{-\dfrac{\pi}{2}+2\pi(1)}{4}))

=23/4(cos(3π8)+isin(3π8))=2^{3/4}(\cos(\dfrac{3\pi}{8})+i\sin(\dfrac{3\pi}{8}))

=21/422+i21/42+2=2^{-1/4}\sqrt{2-\sqrt{2}}+i2^{-1/4}\sqrt{2+\sqrt{2}}

k=2:k=2:

84(cos(π2+2π(2)4)+isin(π2+2π(2)4))\sqrt[4]{8}(\cos(\dfrac{-\dfrac{\pi}{2}+2\pi(2)}{4})+i\sin(\dfrac{-\dfrac{\pi}{2}+2\pi(2)}{4}))

=23/4(cos(7π8)+isin(7π8))=2^{3/4}(\cos(\dfrac{7\pi}{8})+i\sin(\dfrac{7\pi}{8}))

=21/42+2+i21/422=-2^{-1/4}\sqrt{2+\sqrt{2}}+i2^{-1/4}\sqrt{2-\sqrt{2}}

k=3:k=3:

84(cos(π2+2π(3)4)+isin(π2+2π(3)4))\sqrt[4]{8}(\cos(\dfrac{-\dfrac{\pi}{2}+2\pi(3)}{4})+i\sin(\dfrac{-\dfrac{\pi}{2}+2\pi(3)}{4}))

=23/4(cos(11π8)+isin(11π8))=2^{3/4}(\cos(\dfrac{11\pi}{8})+i\sin(\dfrac{11\pi}{8}))

=21/422i21/42+2=-2^{-1/4}\sqrt{2-\sqrt{2}}-i2^{-1/4}\sqrt{2+\sqrt{2}}

2.


cos(4α)+isin(4α)=(cosα+isinα)4\cos(4\alpha)+i\sin(4\alpha)=(\cos \alpha+i\sin \alpha)^4

=cos4α+4icos3αsinα6cos2αsin2α=\cos^4\alpha+4i\cos^3\alpha\sin\alpha-6\cos^2\alpha\sin^2\alpha

4icosαsin3α+sin4α-4i\cos\alpha\sin^3\alpha+\sin^4\alpha

cos(4α)=cos4α6cos2αsin2α+sin4α\cos(4\alpha)=\cos^4\alpha-6\cos^2\alpha\sin^2\alpha+\sin^4\alpha



cos(5α)+isin(5α)=(cosα+isinα)5\cos(5\alpha)+i\sin(5\alpha)=(\cos \alpha+i\sin \alpha)^5

=cos5α+5icos4αsinα10cos3αsin2α=\cos^5\alpha+5i\cos^4\alpha\sin\alpha-10\cos^3\alpha\sin^2\alpha

10icos2αsin3α+5cosαsin4α+isin5α-10i\cos^2\alpha\sin^3\alpha+5\cos \alpha\sin^4\alpha+i\sin^5\alpha

sin(5α)=5cos4αsinα10cos2αsin3α+sin5α\sin(5\alpha)=5\cos^4\alpha\sin\alpha-10\cos^2\alpha\sin^3\alpha+\sin^5\alpha

3,


(2cosα)6=(z+z1)6=z6+6z5z1+15z4z2(2\cos\alpha)^6=(z+z^{-1})^6=z^6+6z^5z^{-1}+15z^4z^{-2}

+20z3z3+15z2z4+6zz5+z6+20z^3z^{-3}+15z^2z^{-4}+6zz^{-5}+z^{-6}

=(z6+z6)+6(z4+z4)+15(z2+z2)+20=(z^6+z^{-6})+6(z^4+z^{-4})+15(z^2+z^{-2})+20

=2cos(6α)+12cos(4α)+30cos(2α)+20=2\cos(6\alpha)+12\cos(4\alpha)+30\cos(2\alpha)+20

cos6α=132cos(6α)+316cos(4α)+1532cos(2α)+516\cos^6\alpha=\dfrac{1}{32}\cos(6\alpha)+\dfrac{3}{16}\cos(4\alpha)+\dfrac{15}{32}\cos(2\alpha)+\dfrac{5}{16}


4.


(2cosα)3=(z+z1)3(2\cos\alpha)^3=(z+z^{-1})^3

(2isinα)4=(zz1)4(2i\sin\alpha)^4=(z-z^{-1})^4

128cos3αsin4α=(z+z1)3(zz1)4128\cos^3\alpha\sin^4\alpha=(z+z^{-1})^3(z-z^{-1})^4

=(z2z2)3(zz1)=(z^2-z^{-2})^3(z-z^{-1})

=(z63z2+3z2z6)(zz1)=(z^6-3z^2+3z^{-2}-z^{-6})(z-z^{-1})

=z7z53z3+3z+3z13z3z5+z7=z^7-z^5-3z^3+3z+3z^{-1}-3z^{-3}-z^{-5}+z^{-7}

=(z7+z7)(z5+z5)3(z3+z3)+3(z+z1)=(z^7+z^{-7})-(z^5+z^{-5})-3(z^3+z^{-3})+3(z+z^{-1})

=2cos(7α)2cos(5α)6cos(3α)+6cosα=2\cos(7\alpha)-2\cos(5\alpha)-6\cos(3\alpha)+6\cos\alpha

cos3αsin4α=164cos(7α)164cos(5α)\cos^3\alpha\sin^4\alpha=\dfrac{1}{64}\cos(7\alpha)-\dfrac{1}{64}\cos(5\alpha)

364cos(3α)+364cosα-\dfrac{3}{64}\cos(3\alpha)+\dfrac{3}{64}\cos\alpha




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