Problem 1 - Precalculus Mathematics for Calculus

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: [-π/2, π/2]. (b) Domain: [-1, 1], Range: [0, π]. (c) Domain: (-∞, ∞), Range: (-π/2, π/2).

Step 1: Explaining Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1} \), is the function that "undoes" the sine function over a specific domain and range. The domain of \( \sin^{-1} \) is the set of values for which the function is defined, and this is the interval \([-1, 1]\). The range is the set of output values, which corresponds to the angles, and for \( \sin^{-1} \), this is \([-\pi/2, \pi/2]\).

Step 2: Explaining Inverse Cosine Function

The inverse cosine function, represented by \( \cos^{-1} \), is defined similarly, but for the cosine function. The domain of \( \cos^{-1} \) is also \([-1, 1]\), as it only takes values within this interval. The range is different; the outputs are angles between \([0, \pi]\).

Step 3: Explaining Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1} \), works with the tangent function's domain and range. Unlike the other two, the domain of \( \tan^{-1} \) is all real numbers, \(( -\infty, \infty )\), because tangent can take any real number as an input. Its range is \((-\pi/2, \pi/2)\), the set of angles \(\theta\) whose tangent values it returns.

Key Concepts

Domain and Range

Understanding the domain and range of inverse trigonometric functions is crucial for working with these functions effectively. The domain is the set of all possible input values for which the function is defined. The range is the set of possible output values the function can produce.

For inverse trigonometric functions:

• Domain represents the set of values the original trigonometric function can never exceed.
• Range identifies the angles or results you will obtain from the inverse function.

These definitions play a significant role in determining how we interpret the inverse trigonometric functions and apply them in mathematical problems.

Inverse Sine Function

The inverse sine function, represented as \( \sin^{-1} \) or \( \arcsin \), is used to find an angle whose sine value gives a specific number.

In terms of domain and range:

• The domain of \( \sin^{-1} \) is \([-1, 1]\). This is because no sine of a real angle can be more than 1 or less than -1.
• The range of \( \sin^{-1} \) is \([-\pi/2, \pi/2]\). This is because the inverse sine covers the entire first and fourth quadrants.

These values ensure that the inverse sine function returns only the principal value or the main angle for a sine function.

Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1} \) or \( \arccos \), helps find an angle whose cosine equals a given number.

For the inverse cosine function:

• The domain is \([-1, 1]\). Like sine, cosine values range between -1 and 1.
• The range of \( \cos^{-1} \) is \([0, \pi]\), covering angles in the first and second quadrants.

This range is crucial as it provides the principal angle, or the most common measurement, for an angle whose cosine has a real value.

Inverse Tangent Function

Inverse tangent, labeled as \( \tan^{-1} \) or \( \arctan \), calculates the angle whose tangent results in a specified value.

In terms of domain and range, \( \tan^{-1} \) showcases:

• A domain of \(( -\infty, \infty )\). Tangent values are not bounded, which means any real number is a possible input.
• A range of \((-\pi/2, \pi/2)\). This range is based on the fact that tangent functions repeat every \(\pi\) radians, but for \( \tan^{-1} \), it focuses on angles from the first and fourth quadrants.

The inverse tangent is unique in allowing complete functionality over all real numbers, making it particularly versatile for applications.