Problem 1 - Algebra and Trigonometry

Question

The inverse sine, inverse cosine, and inverse tangent functions have the followings domains and ranges. (a) The function \(\sin ^{-1}\) has domain ________ and range _________ (b) The function \(\cos ^{-1}\) has domain ________ and range ________ (c) The function \(\tan ^{-1}\) has domain ________ and range _________

Step-by-Step Solution

Answer

(a) Domain: [-1,1], Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\); (b) Domain: [-1,1], Range: [0, \(\pi\)]; (c) Domain: \((-\infty, \infty)\), Range: \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 1: Understanding Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is the inverse of the sine function when restricted to its principal interval. Therefore, its domain is the range of the sine function for \( x \) values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which is \([-1, 1]\). Its range is the output values which correspond to these inputs, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Understanding Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1}(x) \) or \( \arccos(x) \), is the inverse of the cosine function in the interval where \( x \) lies in \([0, \pi]\). Thus, its domain is the range of the cosine function on \([0, \pi]\), which is \([-1, 1]\), and the range is \([0, \pi]\) because those are the values \( \cos^{-1} \) can yield.

Step 3: Understanding Inverse Tangent Function

The inverse tangent function, expressed as \( \tan^{-1}(x) \) or \( \arctan(x) \), stems from the tangent function's restricted domain to \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). Hence, the domain is all real numbers \((-\infty, \infty)\) because tangent can take any real input within its range, and the range reflects the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\).

Key Concepts

Inverse Sine Function

The inverse sine function, often denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is quite fascinating as it reverses the sine function's role when restricted appropriately. The domain of the inverse sine function is the set of all inputs that the sine function can produce outputs for, which is

• [-1, 1]

This means that only values within this interval can be input into \( \sin^{-1} \). Now, let's talk about its range: the possible outputs. The range of the inverse sine function is the range of angles it "maps back" to, which is

• [\(-\frac{\pi}{2}, \frac{\pi}{2}\)]

This indicates that \(\arcsin(x)\) generates angles in radians that lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). These outputs conform to the limits that mirror the sine function's peak and valley on the unit circle.
Using the inverse sine is quite helpful in solving trigonometric equations where the unknown angle needs to be found from a specific sine value.

Inverse Cosine Function

Similarly, the inverse cosine function, denoted \( \cos^{-1}(x) \) or \( \arccos(x) \), acts inversely to the cosine function but only within a particular interval. To understand how this works, first consider its domain:

• [-1, 1]

This indicates that only values from -1 to 1, which encompass every possible output of the cosine function, can be inputs for \( \cos^{-1} \). For its range, the set of angles \( \cos^{-1}(x) \) can emit, think about how it correlates to the cosine curve's behavior over a particular spectrum of angles:

• [0, \(\pi\)]

Using radians, \(\arccos(x)\) provides outputs between 0 and \(\pi\). This span reflects a key area where the cosine curve typically swings, covering a full half-turn from 0 to \(\pi\) radians on the unit circle. Understanding this can help immensely in cases where the cosine angle is known, and the measure of that angle is required.

Inverse Tangent Function

The inverse tangent function, noted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is a bit different since the tangent function can take on a very broad scope of values considering the nature of the tangent line. First, let's review the domain:

• All real numbers \((-\infty, \infty)\)

This signifies that unlike the inverse sine and cosine functions, \(\arctan(x)\) can accept any real number as its input. The inverse tangent function is quite flexible this way. It's equally important to grasp the concept of its range:

• \((-\frac{\pi}{2}, \frac{\pi}{2})\)

Here, the range shows that \(\tan^{-1}(x)\) delivers outputs that hover between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, but importantly, these do not include the endpoint angles since the tangent line approaches, but never reaches, these vertical asymptotes in its curve.
The versatility of the \(\arctan(x)\) function makes it useful in many scenarios where a wide array of real number solutions is needed.

Domain and Range

When discussing inverse trigonometric functions, understanding the terms "domain" and "range" is crucial. Each function has specific domains and ranges, defining what inputs it can accept and what outputs it could produce.

• Domain: The set of all possible input values of a function.
• Range: The set of all possible output values a function can produce.

In the case of inverse functions:

• For \(\sin^{-1}(x)\) and \(\cos^{-1}(x)\), the domain is \([-1, 1]\) because sine and cosine outputs never go beyond these extremes.
• The range for \(\sin^{-1}(x)\) and \(\tan^{-1}(x\)) is \([\(-\frac{\pi}{2}, \frac{\pi}{2}\)\) as these values represent the principal range of angles for these functions.
• The range for \(\cos^{-1}(x)\) is \([0, \pi]\), as this range accommodates the inverse cosine values needed to map original cosine inputs correctly.

Understanding these foundational ideas of domain and range allows for efficient problem-solving when handling inverse functions.

Arc Functions

Inverse trigonometric functions are also known as "arc functions", which is pivotal in contexts involving angles and radians. The term "arc" comes from how these functions originate from interpreting the circumference of a circle, specifically the arc's angular span.

• Arc Names: These functions may also be called \(\arcsin, \arccos,\) and \(\arctan\).
• Radians Interpretation: They typically provide angles in radians, a natural unit of measurement in circle-related math.
• Real-world Application: Used keenly in fields requiring precision and control involving waves, rotations, and angles.

By interpreting angles based on arc measures, arc functions translate complex trigonometric identities into simpler angular terms, paving the way for clearer and more intuitive understanding. This interplay of trigonometric principles aids in transforming coordinate data and solving complex equations, spanning mathematics, physics, and beyond.
Arc functions thus hold a remarkable position in mathematical analysis and everyday calculations related to spherical data and periodic functions.