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Magazine Chapter 1: Problem 12
Sketch the graph of the given equation in the complex plane. $$ \arg (z)=\pi / 4 $$
Short Answer
Expert verified
The graph is a 45-degree line through the origin in the complex plane.
Step by step solution
01
Understand the Argument Function
The argument \(\arg(z)\) of a complex number \(z = x + yi\) is the angle \(\theta\) made with the positive real axis. Here, you are given that \(\arg(z) = \frac{\pi}{4}\). This means that the angle that the line, connecting the origin (0,0) to the point (x,y), makes with the positive x-axis is \(\frac{\pi}{4}\) radians, or 45 degrees.
02
Relate Argument to Cartesian Coordinates
Since the argument is \(\frac{\pi}{4}\), the line where this is true corresponds to points where \((x,y)\) is such that the angle from the positive x-axis to the point \(x + yi\) is 45 degrees. In trigonometric terms, \((x,y)\) satisfies: \( an(\theta) = \frac{y}{x} = 1\) because \(\tan(\frac{\pi}{4}) = 1\). This gives us the line equation \(y = x\) in the Cartesian plane.
03
Sketch the Graph on the Complex Plane
To sketch the graph of this equation, draw a line through the origin with a slope of 1 (i.e., a 45-degree line). This line will angle upwards to the right, crossing through points like (1,1), (2,2), (-1,-1), etc. This line represents all complex numbers z that have an argument of \(\frac{\pi}{4}\).
04
Include Both Directions
Remember, the argument can be positive or negative based on how it's measured, but for \(\frac{\pi}{4}\), it's always the same 45-degree line through the origin. This implies that the line extends infinitely in both directions, encompassing all complex numbers whose angle from the positive x-axis is 45 degrees.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Argument of a Complex Number
The argument of a complex number is a fundamental concept used to describe the position of that number in the complex plane. When we say the argument of a complex number, we're actually referring to the angle between the positive real axis and the line connecting the origin to the point representing the complex number.
- This angle is often denoted as \( \theta \) and measured in radians.
- The argument, \( \arg(z) \), is defined for a complex number \( z = x + yi \).
- For our specific problem, we're given that \( \arg(z) = \frac{\pi}{4} \).
Trigonometry in Complex Numbers
Trigonometry plays a key role in understanding complex numbers, especially when working with the argument. The trigonometric functions help relate the angles to the coordinates of the points in the complex plane.
- The tangent of an angle \( \theta \) in the context of complex numbers can be determined using the ratio of the imaginary part to the real part, \( \tan(\theta) = \frac{y}{x} \).
- In our case, because \( \arg(z) = \frac{\pi}{4} \) or 45 degrees, we have \( \tan(\frac{\pi}{4}) = 1 \).
- This condition implies that \( y = x \) for any point \( (x, y) \) on this line.
Graphing in the Complex Plane
Graphing complex numbers involves plotting them on a plane where the x-axis represents the real part and the y-axis represents the imaginary part. Visualizing these numbers can give us intuitive insights into their nature and relationships.
- In our example, plotting all complex numbers \( z \) with \( \arg(z) = \frac{\pi}{4} \) involves drawing the line \( y = x \).
- This means any point on this line can be considered as \( x + xi \), where \( x \) is any real number.
- The line through the origin forms a perfect 45-degree angle with the real axis.
