设复数z满足|arg((z+1)/(z+2))|=π/6, 求argz的取值范围

Translating the analysis

I'm translating the problem's argument description from Chinese, identifying key relationships between terms like | (z + 1) / (z + 2) | and setting w = z + 2.

Establishing relationships

I’m clarifying that w represents a shift of z by 2 in the real axis' positive direction. This makes me think about vectors and their implications.

Imagining the shift

OK, let's think about how shifting w by 1 to the right gives w - 1. This helps me see the ± π/6 angle between these vectors from the origin in the complex plane.

Re-imagining z

Mapping out z in polar form, I’m noting that the locus forms an angle of ± π/6 between vectors from z to -1 and z to -2.

Piecing it together

Interestingly, I'm connecting the argument difference to the angle between lines from z to -1 and -2, simplifying the geometry interpretation.

Taking a closer look

I'm examining how the argument difference between (z+1)/(z+2) translates to the angle between vectors from the origin to z+1 and z+2, simplifying the geometric interpretation.

Determining the range

Piecing together the geometry, I'm focusing on how the angle ± π/6 between vectors (z+2) and (z+1) affects the range of arg z in the complex plane.

Charting new territory

I’m working through geometrically defining arg z by examining the angle ± π/6 between vectors extending from the origin to z+1 and z+2. Progress is steady and promising.

Shifting perspectives

By renaming w as z and redefining the coordinate system, the problem simplifies to finding the range of arg z, interpreted through the new coordinates.

Finding the argument

I’m focusing on the angle of ± π/6 between vectors w and w-1, aiming to determine possible values of arg(w - 2) using geometric interpretations.

Analyzing vector angles

I’m considering the vectors w and w-1, horizontally separated by 1, and examining how the ± π/6 angle between them impacts the argument of w in the complex plane.

Tracing the path

I’m exploring the geometric locus of w, noting it lies on a circle through 0 and 1, with the constant angle of ± π/6 subtended at w from 0 and 1.

Charting the angle

OK, let’s explore this: mapping out the angle of ± π/6 between vectors from the origin to w and w-1, revealing insights into w's position in the plane.

Analyzing vector angles

I’m using dot product calculations to determine the angle between vectors (x,y) and (x-1,y), focusing on the ± π/6 angle difference.

Analyzing vector relationships

I’m calculating the dot product and magnitudes of vectors (x,y) and (x-1,y) to determine the range of arg((x-2) + i y) that satisfies the ± π/6 angle condition.

Wondering about geometry

I’m piecing together how the constant angle between lines to (z+1) and (z+2) might connect to a circular locus or specific geometric relationships in the complex plane.

Mapping out the region

I'm considering how, as r approaches infinity, the difference in arguments for points z+1 and z+2 stays constant, potentially highlighting a specific area in the complex plane.

Revisiting the challenge

Let me see. By focusing on the relationship between z+1 and z+2, the constant angle of ± π/6 can define a specific region in the complex plane.

Analyzing angle differences

I’m piecing together the angle from the dot product of vectors (r cosθ + 1, r sinθ) and (r cosθ + 2, r sinθ), aiming for a simplified geometric interpretation.

Pulling together

By incorporating the magnitudes of vectors A and B, I’m using the dot product to relate the angle difference to cos(π/6). This approach helps refine the geometric interpretation.

Solving and transforming

I’m breaking down the equation by squaring both sides and cross-multiplying, leading to expanding and transforming terms to find the range of arg((x-2) + i y) under the constant angle condition.

Breaking down the equation

I'm expanding both sides by substituting and simplifying terms to capture the range of arg((x-2) + i y) that meets the constant angle condition.

Combining terms

I’m expanding A and B by multiplying their terms, constructing a polynomial expression that hints at the geometric relationship between angles and distances in the complex plane.

Combining terms

I’ve simplified the equation by merging like terms, focusing on the r^4, r^3c, r^2, rc, and constant terms, aiming to find a simplified form.

Verifying the solution

I'm checking if the expression can be zero for real r>0, considering the angle difference of ± π/6 and simplifying the equation to confirm the geometric interpretation.

Reassessing circular arc theorem

I’m thinking through the theorem that links the constant angle between lines from the origin to fixed points to a circular arc through those points, rechecking the geometric interpretation.

Revisiting the geometry

I’m realizing the points (z+1) and (z+2) shift as z changes, which complicates their fixed-position interpretation. This dynamic alignment needs careful geometric consideration.

Taking a closer look

I'm focusing on how z's argument can be confined to a sector near the negative real axis, ensuring the angle difference of ± π/6 is preserved.

Thinking through constraints

When the angle between z+1 and z+2 is restricted to π/6, it naturally places z within a sector, particularly near the negative real axis.

Refining the analysis

OK, let me see. I’m considering z on the imaginary axis and calculating the angle difference between vectors z+1 and z+2 to pinpoint the sector.

Mapping the angle

I'm exploring the range of z's argument, considering symmetry around the negative real axis, and narrowing down to predictions like arg z in [-π/3, π/3].

Testing different z positions

I’m checking z’s influence on angle differences, using special cases like pure real, pure negative, and positive imaginary positions to see if they meet the criteria.

Understanding symmetry

Piecing together the equality of argument differences being ± π/6, I’m thinking about the geometric implications and how this symmetry defines the locus of points z.

Identifying the locus

I’m examining the locus of z by considering if it could be a circle or arc, and whether the argument of z must stay within a certain range to satisfy the condition.

Testing the hypothesis

Using the tangent formula, I’m taking a closer look at whether the equation has real solutions for y by checking the discriminant, which gives a negative and confirms no real solutions.

Examining the boundary

Interestingly enough, no real solutions appear for y on the positive or negative imaginary axis, suggesting the boundary might reside away from the imaginary axis.

Evaluating conditions

I’m determining if there are solutions when z is less than -2, ensuring the argument difference is ± π/6. So far, no solutions seem apparent.

Piecing it together

OK, I’m seeing that real numbers, especially those between -2 and -1, don’t work since they fail to meet the ± π/6 angle difference.

Navigating angle restrictions

I’m thinking through how real z values, especially those between -2 and -1, fall short of the ± π/6 angle difference, hinting that z needs an imaginary component.

Demonstrating angle differences

By working through the complex plane, I’m defining z's real and imaginary parts, examining their angles, and showing that they meet the ± π/6 difference condition.

Examining transformations

I'm investigating how transforming z into t by dividing (z+1) by (z+2) with an argument of ± π/6 affects the argument of t, yielding specific complex values.

Mapping the transformation

OK, let me see. By transforming z into t and back, I’m identifying z's as points on a specific path in the complex plane, corresponding to t's values.

Mapping out the curve

OK, let me see: as $\rho$ changes from 0 to $\infty$, $\rho e^{i(\pm \pi/6)}$ traces a path in the z-plane, likely forming a circle or a line.

Tracing transformations

Analyzing extreme values of $\rho$, I aim to uncover the path set by $\rho e^{i(\pm \pi/6)}$, ultimately establishing $z$'s trajectory.

Analyzing $z$ transformations

I’m examining how $\rho e^{i(\pm \pi/6)}$ maps to $z$, determining that $z$ moves between $-1$ and $-2$ on the negative real axis as $\rho$ transitions from 0 to $\infty$.

Mapping transformations

Piecing together the map from $\rho e^{i(\pm \pi/6)}$ to $z$, it seems $z$ traces a path between $-1$ and $-2$ on the negative real axis.

Analyzing boundary conditions

Testing the lower and upper half-planes to help pinpoint $z$'s location and clarify if the boundary is within these regions. Progressing towards a solution.

Evaluating specific parameters

I’m thinking through the implications of setting $r = 1$, working out $z(1)$ using $e^{i\pi/6}$ to figure out $z$'s behavior in the complex plane.

Working out the details

I’m multiplying the numerator and denominator by the conjugate of the denominator, then breaking down the real and imaginary parts to find $z(1)$. This clarifies the result nicely.

Breaking down expressions

I’m separating and simplifying each term to calculate the real part of the denominator, using terms like $(\sqrt{3}-1)$ and $(1 - \sqrt{3}/2)$.

Piecing together

I’m adding the imaginary parts of the numerator and denominator of $z(1)$, simplifying to find $z(1)$ as $[ [ 3(\sqrt{3}-2 )/2 ] + i(1/2 ) ] / ( 2- \sqrt{3} )$.

Breaking down the pieces

OK, let me see: after rationalizing the denominator and distributing, $z(1)$ simplifies to $-3/2 + i( (2+ \sqrt{3})/2 )$. This really clarifies things.

Cracking the argument

Taking a closer look at $z(1)$ in the second quadrant, I covert the coordinates to an angle of 2.2456 radians, which gives a clear indication of the argument.

Piecing together

I'm working through the range of $\text{arg}(z)$ by analyzing how the real and imaginary parts of $z$ shift with $r$ to confirm the argument’s span.

Pinpointing the range

Taking a closer look, the argument of $z$ is likely between $\frac{5\pi}{6}$ and $\frac{7\pi}{6}$, symmetrical around $\pi$.

Concluding the range

I’m concluding that the argument of $z$ spans from $\frac{2\pi}{3}$ to $\frac{4\pi}{3}$, encompassing the entire sector around the negative real axis.

Assessing argument validity

I'm examining the argument of $z$ and its movements between 120 and 240 degrees. This makes me think the sector range could be possible for all $r$.

Plotting the course

I’m concluding that the argument of $z$ spans from $\frac{2\pi}{3}$ to $\frac{4\pi}{3}$, which covers the entire sector around the negative real axis.

Examining the argument

Taking a closer look, the argument of $z$ spans from 128 degrees to 180 degrees as $r$ ranges from 0 to 1. This narrows down the range effectively.

Analyzing the range

Mapping out the argument of $z$ as $r$ spans from 128 degrees to 180 degrees, I’m beginning to see a defined range in radians.

Examining argument nuances

I'm working through the symmetrical range for $\phi = -\pi/6$ in the lower half-plane. The principal argument union seems to be $[ -\pi, -2.234... ]$ union $[ 2.234..., \pi ]$.

Streamlining the analysis

I’m simplifying the argument to arctan of a ratio of imaginary to real parts, revealing the angle is around 128.6 degrees, confirming the range as $[ 2.23, \pi ]$ union $(-\pi, -2.23]$.

Piecing together the interval

OK, let me see. The range $[\frac{5\pi}{6}, \pi) \cup (-\pi, -\frac{5\pi}{6}]$ is intriguing, but it doesn't include the angle around 128 degrees.

Confirming the range

I'm working through the argument range for $z$, specifically around 128 degrees, and finding it fits within $(\frac{2\pi}{3}, \frac{4\pi}{3})$.

Confirming inclusion

I'm examining if $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ are within the range. For $r=0$, $z= -1$, and $\pi$ is clearly inside.

Verifying range inclusion

I’m checking if $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ are included. Imaginary/real ratio is negative, consistent with quadrant 2 and 3. Progress is steady.

Confirming final boundary

I’m thinking through the region around the negative real axis, excluding the positive real axis. The final result places $\text{arg}(z)$ between $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$.

Thought for 12m 49s