The polar form of
𝑧
=
−
6
7
+
3
2
7
𝑖
is:

Hint:
𝑧
=
𝑎
+
𝑏
𝑖
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
where
𝑟
=
𝑎
2
+
𝑏
2
and
𝜃
=
tan
−
1
⁡
𝑏
𝑎
.

Also, don't forget to plot the complex number on the Argand diagram.

Question

The polar form of  
𝑧
=
−
6
7
+
3
2
7
𝑖
is:

Hint: 
𝑧
=
𝑎
+
𝑏
𝑖
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
 where  
𝑟
=
𝑎
2
+
𝑏
2
  and  
𝜃
=
tan
−
1
⁡
𝑏
𝑎
.

Also, don't forget to plot the complex number on the Argand diagram.

BlackTom AI · Accepted Answer

The question as provided asks for the polar form of z, with a hint that z = a + b i = r (cos θ + i sin θ), where r = sqrt(a^2 + b^2) and θ = arctan(b/a). It also mentions plotting the point on the Argand diagram. However, there is a significant formatting problem in the input: the complex number z is written as "− 6 7 + 3 2 7 i" which is not a standard or parseable representation of a + bi, and the answer options section is empty, making it impossible to identify what choices (if any) we should evaluate.

First, a general step-by-step approach to converting a complex number to polar form, which is the method you would apply once a and b are clearly identified, is as follows:
- Identify a and b from z = a + b i. If the number is real and/or imaginary, write down the real part a and the imaginary part b explicitly.
- Compute r = sqrt(a^2 + b^2). This r is the modulus of z and serves as the radial component in polar form.
- Compute θ = atan2(b, a). The function atan2 correctly handles the quadrant and gives the argument (angle) of z with respect to the positive real axis.
- Write z in polar form as z = r [cos(θ) + i sin(θ)], or equivalently z = r cis(θ).
- If desired, you can express θ in terms of radians, for example in multiples of π (e.g., θ = 2π/3) when the ratio b/a corresponds to a notable angle.

Next, about the provided answer candidates: since the "answer_options" field is empty, there are no specific alternatives to analyze for correctness or incorrectness. Without explicit options, we cannot perform a choice-by-choice evaluation or explain why one option is right and others are wrong.

Additionally, the sole available "answer" entry appears to be a single polar-form-like expression, but due to the broken formatting it is unclear which part represents the modulus r, the angle θ, or even the intended numeric values. Examples of a well-formed polar form would be something like z = 4 (cos(π/3) + i sin(π/3)) or z = 4 cis(π/3), where r = 4 and θ = π/3. The fragment shown ("2 6 7 ( cos 2π/3 + i sin 2π/3 )") seems to be attempting r = 2/7 in combination with θ = 2π/3, but this is speculative and cannot be treated as a verified correct answer without proper formatting and context.

In a properly formatted version, you would:
- Confirm the numeric real part a and imaginary part b of z.
- Compute r = sqrt(a^2 + b^2).
- Determine θ = atan2(b, a).
- Present z in the required polar form and verify it by expanding r cos θ and r sin θ to recover a and b.

Because the essential data (the clean a and b values and the list of answer options) is missing or garbled, the analysis cannot proceed to identify which option is correct or incorrect, nor can it justify choices. If you can provide a clean version of the complex number z and, if applicable, the complete set of answer options, I can give a detailed, option-by-option explanation that differentiates correct reasoning from common misconceptions.

SMAT011 Weekly Quiz 3 |LA003

The polar form of 𝑧 = − 6 7 + 3 2 7 𝑖 is: Hint: 𝑧 = 𝑎 + 𝑏 𝑖 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) where 𝑟 = 𝑎 2 + 𝑏 2 and 𝜃 = tan − 1 ⁡ 𝑏 𝑎 . Also, don't forget to plot the complex number on the Argand diagram.

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