中文
Find the argument of $-4+5 i$ in the range $[0,2 \pi)$. (Correct the answer to 2 decimal places.)

Answer: $\square$

Question

中文
Find the argument of $-4+5 i$ in the range $[0,2 \pi)$. (Correct the answer to 2 decimal places.)

Answer: $\square$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the argument of the complex number $-4 + 5i$, we use the formula for the argument: $ heta = an^{-1}\left(\frac{b}{a} ight)$, where $a = -4$ and $b = 5$. ‖ ‖ step 2 ⋮ Substituting the values into the formula gives us: $ heta = an^{-1}\left(\frac{5}{-4} ight)$. ‖ ‖ step 3 ⋮ Since the point $(-4, 5)$ is in the second quadrant, we need to add $\pi$ to the angle obtained from the arctangent function. Thus, $ heta = an^{-1}\left(\frac{5}{-4} ight) + \pi$. ‖ ‖ step 4 ⋮ Calculating this gives us approximately $ heta \approx 2.4981$. Rounding to two decimal places gives us: $ heta \approx 2.50$. ‖ ∻Answer∻ ⚹ 2.50 ⚹ ∻Key Concept∻ ⚹The argument of a complex number is the angle formed with the positive real axis in the complex plane.⚹ ∻Explanation∻ ⚹The argument is calculated using the arctangent function, and adjustments are made based on the quadrant in which the complex number lies. In this case, the complex number $-4 + 5i$ is in the second quadrant, leading to the final argument of approximately 2.50.⚹◊What is the relationship between the argument and the modulus of a complex number?⍭ Generate me a similar question◊

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