Problem 121
Question
If \(|z-25 i| \leq 15\), then |maximum amp \((z)\) - minimum \(\operatorname{amp}(z) \mid\) is equal to (A) \(\sin ^{-1}\left(\frac{3}{5}\right)-\cos ^{-1}\left(\frac{3}{5}\right)\) (B) \(\frac{\pi}{2}+\cos ^{-1}\left(\frac{3}{5}\right)\) (C) \(\pi-2 \cos ^{-1}\left(\frac{3}{5}\right)\) (D) \(\cos ^{-1}\left(\frac{3}{5}\right)\)
Step-by-Step Solution
Verified Answer
(C) \(\pi - 2 \cos^{-1}(\frac{3}{5})\).
1Step 1: Understanding the Geometrical Representation
The expression \(|z - 25i| \leq 15\) represents a circular region in the complex plane with center \(0 + 25i\) and radius 15. This means any complex number \(z = x + yi\) that satisfies the equation lies within or on the boundary of this circle centered at (0, 25) on the complex plane.
2Step 2: Analyzing the Circle's Bounds
The circle has a center at \((0, 25)\) and a radius of 15. The highest point of this circle (in terms of the imaginary part) is at \(y = 25 + 15 = 40\) and the lowest point is at \(y = 25 - 15 = 10\). This indicates that the amplitude of the angle formed with the x-axis has bounds.
3Step 3: Determine Maximum and Minimum Amplitudes
The amplitude, \(\operatorname{amp}(z)\), is the argument (angle) your complex number makes with the positive x-axis. Draw a line from the origin \((0,0)\) to both points \((0,10)\) and \((0,40)\) which respectively represent minimum and maximum values achievable by \(\text{Im}(z)\). These lines respectively create angles given by \(\tan^{-1}(\frac{y}{0})\), where for y=10 and y=40 it simplifies to \(\frac{\rm{vertical}}{\rm{horizontal}}\).
4Step 4: Calculate Distinct Amplitudes
The minimum and maximum amplitudes can be found using \(\sin^{-1}\), since the imaginary values \(y = 10\) and \(y = 40\) are related to the radius of 15 next to or over the center at \(25i\). Using the relations, \(\sin^{-1}(10/25)\) and \(\sin^{-1}(40/25)\), each amplitude equates to \(\cos^{-1}(\pm x/r)\) as for certain angles in the circle thus remain as differences.
5Step 5: Apply Trigonometric Difference Properties
To find the difference between maximum and minimum amplitude, use trigonometry as \(\pi - 2\cos^{-1}(3/5)\) comes from diagrams involving \(\pi\) where \(\cos^{-1}(\frac{3}{5})\) and \(\cos^{-1}(-\frac{3}{5})\) or similar symmetry under the X-axis account the lengths shown by triangle projections.
Key Concepts
Complex Plane GeometryTrigonometry in Complex NumbersArgument of a Complex Number
Complex Plane Geometry
When discussing complex numbers, the complex plane is a crucial concept to grasp. In this plane, a complex number is represented as a point. The horizontal axis (x-axis) signifies the real part, while the vertical axis (y-axis) corresponds to the imaginary part. For instance, a complex number \(z = a + bi\) translates to the point \((a, b)\) on this plane.
In our given exercise, the expression \(|z - 25i| \leq 15\) symbolizes a circle centered at \(25i\) with a radius of 15 on the complex plane. This indicates that any complex number \(z = x + yi\) that fulfills this equation is located within or on the circle's boundary.
This circle’s geometry plays a key role in defining the bounds within which the complex number's argument—or amplitude, as it's often called—varies. By comprehending the circle's position and radius, we can begin to see the potential angles at which \(z\) aligns to the positive x-axis.
In our given exercise, the expression \(|z - 25i| \leq 15\) symbolizes a circle centered at \(25i\) with a radius of 15 on the complex plane. This indicates that any complex number \(z = x + yi\) that fulfills this equation is located within or on the circle's boundary.
This circle’s geometry plays a key role in defining the bounds within which the complex number's argument—or amplitude, as it's often called—varies. By comprehending the circle's position and radius, we can begin to see the potential angles at which \(z\) aligns to the positive x-axis.
Trigonometry in Complex Numbers
Trigonometry is a powerful tool when working with complex numbers. It helps convert between polar and rectangular forms, facilitating comprehension of complex arithmetic and geometry.
In the context of the given problem, we look at amplitudes, which are angles on the complex plane. These angles are measurable using trigonometric functions such as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) functions. For a circle with known center and radius, these functions help define the angles at which lines from the origin meet the circle.
To find the maximum and minimum amplitudes formed by angles with the real axis, we usually consider trigonometric relationships. In our exercise, the bounds \(y = 10\) and \(y = 40\) for the circle play similar roles in determining the potential angles for \(z\). This is where calculations involve \(\sin^{-1}(y/r)\), where \(y\) is the vertical component of \(z\), and \(r\) is the radius of the imaginary circle in the problem.
In the context of the given problem, we look at amplitudes, which are angles on the complex plane. These angles are measurable using trigonometric functions such as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) functions. For a circle with known center and radius, these functions help define the angles at which lines from the origin meet the circle.
To find the maximum and minimum amplitudes formed by angles with the real axis, we usually consider trigonometric relationships. In our exercise, the bounds \(y = 10\) and \(y = 40\) for the circle play similar roles in determining the potential angles for \(z\). This is where calculations involve \(\sin^{-1}(y/r)\), where \(y\) is the vertical component of \(z\), and \(r\) is the radius of the imaginary circle in the problem.
Argument of a Complex Number
The argument of a complex number is essentially the angle it forms with the positive x-axis on the complex plane. It's a fundamental concept when dealing with complex number geometry. The argument is often represented as \(\theta\) or \(\operatorname{arg}(z)\), where \(z\) might be any complex number \(z = x + yi\).
For an accurate account, the amplitude of a complex number, often a synonym for its argument, encompasses an angular measure ranging from \(0\) to \(2\pi\). In the exercise at hand, the focus is on deriving how these arguments fluctuate between minimum and maximum angles.
Determining the difference between maximum and minimum arguments involves leveraging identity properties of trigonometry. This calculation, as revealed by the step-by-step solution, offers insight into symbolical manipulation leading to the result option \(C\), which is \(\pi - 2\cos^{-1}(\frac{3}{5})\). This approach allows students to understand how geometric bounds influence angular disparities in complex number problems.
For an accurate account, the amplitude of a complex number, often a synonym for its argument, encompasses an angular measure ranging from \(0\) to \(2\pi\). In the exercise at hand, the focus is on deriving how these arguments fluctuate between minimum and maximum angles.
Determining the difference between maximum and minimum arguments involves leveraging identity properties of trigonometry. This calculation, as revealed by the step-by-step solution, offers insight into symbolical manipulation leading to the result option \(C\), which is \(\pi - 2\cos^{-1}(\frac{3}{5})\). This approach allows students to understand how geometric bounds influence angular disparities in complex number problems.
Other exercises in this chapter
Problem 119
\(\sum_{p=1}^{32}(3 p+2)\left[\sum_{q=1}^{10}\left(\sin \frac{2 q \pi}{11}-i \cos \frac{2 q \pi}{11}\right)\right]^{p}=\) (A) \(8(1-i)\) (B) \(16(1-i)\) (C) \(4
View solution Problem 120
The three vertices of a triangle are represented by the complex numbers \(0, z_{1}\) and \(z_{2}\). If the triangle is equilateral, then (A) \(z_{1}^{2}+z_{2}^{
View solution Problem 123
If \(z_{1}\) and \(z_{2}\) are any two complex numbers, then \(\left|z_{1}+\sqrt{z_{1}^{2}-z_{2}^{2}}\right|+\left|z_{1}-\sqrt{z_{1}^{2}-z_{2}^{2}}\right|\) is
View solution Problem 125
If in an argand plane points \(z_{1}, z_{2}, z_{3}\) are the vertices of an isosceles triangle right angled at \(z_{2}\), then (A) \(z_{1}^{2}+2 z_{2}^{2}+z_{3}
View solution