d4nte asked in Science & MathematicsMathematics ยท 2 months ago

Convert Complex Numbers to Rectangular Form.?

How do I continue with 8a? I stopped at 2(cos(pi/4)+isin(pi/4)). But then, I dont understand how to continue afterward.

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  • 2 months ago

    Hello,

    The required knowledge:

    ย ย  ๐‘’โฑแถฟ = cos(๐œƒ) + ๐‘–ยทsin(๐œƒ)

    ย ย  cos(-๐œƒ) = cos(๐œƒ)

    ย ย  sin(-๐œƒ) = -sin(๐œƒ)

    ย ย  cos(๐œ‹/4) = sin(๐œ‹/4) = (โˆš2)/2

    ย ย  cos(๐œ‹/3) = ยฝ

    ย ย  sin(๐œ‹/3) = (โˆš3)/2

    From this, you get:

    2ยท๐‘’^(๐‘–๐œ‹/4)

    ย ย  = 2ยท[cos(๐œ‹/4) + ๐‘–ยทsin(๐œ‹/4)]

    ย ย  = 2ยท[(โˆš2)/2 + ๐‘–ยท(โˆš2)/2]

    ย ย  = (โˆš2) + ๐‘–ยท(โˆš2)ย ย ย  โ—„โ—„โ—„(a)

    5ยท๐‘’^(๐‘–๐œ‹/3)

    ย ย  = 5ยท[cos(๐œ‹/3) + ๐‘–ยทsin(๐œ‹/3)]

    ย ย  = 5ยท[ยฝ + ๐‘–ยท(โˆš3)/2]

    ย ย  = (5/2) + 5๐‘–ยท(โˆš3)/2ย ย ย  โ—„โ—„โ—„(b)

    [๐‘’^(-๐‘–๐œ‹/3)] / 3

    ย ย  = [cos(-๐œ‹/3) + ๐‘–ยทsin(-๐œ‹/3)] / 3

    ย ย  = [cos(๐œ‹/3) โˆ’ ๐‘–ยทsin(๐œ‹/3)] / 3

    ย ย  = [ยฝ โˆ’ ๐‘–ยท(โˆš3)/2] / 3

    ย ย  = (1/6) โˆ’ ๐‘–ยท(โˆš3)/6ย ย ย  โ—„โ—„โ—„(c)

    ๐œ‹ยท๐‘’^(-๐‘–๐œ‹/4)ย ย  = ๐œ‹ยท[cos(-๐œ‹/4) + ๐‘–ยทsin(-๐œ‹/4)]

    ย ย  = ๐œ‹ยท[cos(๐œ‹/4) โˆ’ ๐‘–ยทsin(๐œ‹/4)]

    ย ย  = ๐œ‹ยท[(โˆš2)/2 โˆ’ ๐‘–ยท(โˆš2)/2]

    ย ย  = ๐œ‹(โˆš2)/2 โˆ’ ๐‘–ยท๐œ‹(โˆš2)/2ย ย  โ—„โ—„โ—„(d)

    Regards,

    Dragon.Jade :-)

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  • MyRank
    Lv 6
    2 months ago

    2(cos(ฯ€/4)+isin(ฯ€/4))

    e^(iฮธ) = cos ฮธ + i sin ฮธ

    where ฮธ = ฯ€/4, r = 2

    2(cos(ฯ€/4)+isin(ฯ€/4)) = 2(e^(ฯ€/4i))

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  • 2 months ago

    r e^(iฮธ) = r cis(ฮธ) = r cosฮธ + i r sinฮธ

    cos(ฯ€/4) = sin(ฯ€/4) = โˆšยฝ, so (a) 2 e^(i ฯ€/4) = 2โˆšยฝ + 2โˆšยฝ i = โˆš2 + โˆš2 i

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  • 2 months ago

    Know your unit circle

    cos(ฯ€/4) = โˆš2 / 2

    sin(ฯ€/4) = โˆš2 / 2

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  • 2 months ago

    You win the fried brain award for today.

    a + bi

    a = 2cos(pi/4)

    b = 2sin(pi/4)

    Take a step back. Drink water. Rest

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