Problem 45 - University Calculus: Early Transcendentals

Question

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\tan ^{-1} 2\) b. \(\cos ^{-1} 2\)

Step-by-Step Solution

Answer

\(\tan^{-1} 2\) is defined; \(\cos^{-1} 2\) is not defined.

Step 1: Understand the Notation

The notation \( \tan^{-1} x \) and \( \cos^{-1} x \) represent the inverse trigonometric functions, also known as arc functions. \( \tan^{-1} x \) corresponds to the angle whose tangent is \( x \), and \( \cos^{-1} x \) corresponds to the angle whose cosine is \( x \).

Step 2: Determine the Domain of \( \tan^{-1} x \)

The function \( \tan^{-1} x \) does not have any restrictions on its domain because the tangent function can take any real number as input on the real number line. Therefore, \( \tan^{-1} 2 \) is defined.

Step 3: Determine the Domain of \( \cos^{-1} x \)

The function \( \cos^{-1} x \) is defined for \( x \) values in the range \(-1 \leq x \leq 1\). This is because the cosine of an angle only takes values within this interval. Since \( \cos^{-1} 2 \) has an input value of 2, which is outside this interval, it is not defined.

Key Concepts

Understanding Arctangent

The arctangent function, denoted as \( \tan^{-1} x \), is an inverse trigonometric function. This function helps us find an angle whose tangent value is \( x \).
The tangent function, as seen in trigonometry, can have any real number value. Therefore, when it comes to finding an angle through its arctangent, there is no restriction on the input values for \( x \).
The domain of \( \tan^{-1} x \) includes all real numbers.

• It implies you can input any real number and get a corresponding angle.
• For instance, \( \tan^{-1} 2 \) is clearly defined, because 2 is a real number and lies within the function’s domain.

This characteristic makes arctangent particularly useful in calculating angles from slopes or gradients in geometry or graphing scenarios.

Exploring Arccosine

The arccosine function, represented as \( \cos^{-1} x \), is another type of inverse trigonometric function. It is used to determine an angle whose cosine equals \( x \).
Unlike the arctangent function, the arccosine possesses certain domain restrictions.

• The values of \( x \) must be confined to the range from \(-1 \) to \( 1 \).
• This restriction is due to the nature of the cosine function, which produces outputs only within this interval for angles in the real number system.

So, when you see something like \( \cos^{-1} 2 \), it is considered undefined because 2 is outside the acceptable input range.
The careful consideration of this domain is critical when applying arccosine to solve various problems, such as finding angles in physics and engineering.

Understanding the Domain of Functions

The domain of a function represents all the possible input values for which the function is defined.
Different functions have different domain restrictions based on their mathematical properties.

• For example, polynomial functions generally have an all-encompassing domain that includes all real numbers.
• Trigonometric functions, including their inverses like arccosine, may have limitations on their domains.

Inverse trigonometric functions, in particular, require a careful analysis of their domains:

• The domain of \( \tan^{-1} x \) is all real numbers because the tangent function can output any real value.
• The domain of \( \cos^{-1} x \) is limited to \(-1 \leq x \leq 1\) due to the nature of how the cosine function behaves.

Understanding the domain helps us determine whether a given expression is correctly defined, and it’s an essential skill in solving mathematical problems efficiently.