Anonymous
Anonymous asked in Science & MathematicsMathematics · 8 months ago

Write the complex number in polar form with argument θ between 0 and 2π.?

7+8i

5 Answers

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  • 8 months ago
    Best Answer

    z=a+ib=7+8i

    Polar form

    ρ=√(a²+b²)=√113

    To identify the angle θ, it should be noted that in [0.2π] there are two tangents that take the value a / b.

    The rule states

    • IF a>0 THEN θ=arctan(b/a) which is our case.

    • IF a<0 THEN θ=π+arctan(b/a)

    So il polar form

    7 + 8i=√113 (cos(arctan(8/7) + isin(arctan(8/7))

    Check it out.

    WE have to remember that:

    cos(arctan(x))=1/√(1+x²)

    sin(arctan(x))=x/√(1+x²)

    hence

    cos(arctan(8/7))=1/√(1+(8/7)²)=

    =1/√(113)/7

    sin(arctan(8/7))=x/√(1+(8/7)²)=8/√(113)

    so

    √113(1/√(113)/7+i8/√(113))= 7 + 8i OK!

  • Como
    Lv 7
    8 months ago

    --

    r = 7 + 8 i

    r ² = [ 7² + 8² ] = 49 + 64

    | r | = √113

    tan ∅ = 8/7

    ∅ = 49°

    z = √113 /_ 49°

  • 8 months ago

    Recall:

    z = a + ib ← this is a complex number

    m = √(a² + b²) ← this is its modulus

    tan(α) = b/a → then you can deduce α ← this is the argument

    In your case:

    z = 7 + 8i ← you can see that: a = 7 and you can see that: b = 8

    m = √[(7)² + (8)²]

    m = √(49 + 64)

    m = √113 ← this is the modulus of z

    tan(α) = b/a

    tan(α) = 8/7

    α ≈ 48.814 ° ≈ 0.852 rd

    z = m.[cos(α) + i.sn(α)]

    z = (√113).[cos(0.852) + i.sn(0.852)]

    z = (√113 ; 0.852) ← polar form

  • 8 months ago

    The polar form of a complex number z=a+bi is z=r(cosθ+isinθ), which we might simplify to r cis(θ).

    r = √(7² + 8²) = √113 and θ (in Q1) is arctan(8/7).

    7 + 8i = √113 cis(arctan(8/7))

    You can find approximate values from here if required.

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  • 8 months ago

    r = sqrt(a^2 + b^2) = sqrt(7^2 + 8^2) = sqrt(49 + 64) = sqrt(113) =~ 10.630145812734649407999121914929

    tan(θ) = b/a = 8/7; θ = arctan(8/7) =~ 0.85196632717327211381430754011385

    7 + 8i = r(cos(θ) + i*sin(θ))

    7 + 8i = sqrt(113)*(cos(arctan(8/7)) + i*sin(arctan(8/7)))

    7 + 8i =~ 10.63(cos(0.85) + i*sin(0.85)))

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