Problem 227 - Calculus Volume 1

Question

[T] Use a calculator to evaluate tan \(^{-1}(\tan (2.1))\) and \(\cos ^{-1}(\cos (2.1)) .\) Explain the results of each.

Step-by-Step Solution

Answer

Both expressions return 2.1 because 2.1 lies within the principal range of the inverse functions.

Step 1: Understanding the Expression

We need to evaluate two expressions: 1) \(\tan^{-1}(\tan(2.1))\) and 2) \(\cos^{-1}(\cos(2.1))\). The notation \(\tan^{-1}\) represents the inverse tangent function (arctangent), and \(\cos^{-1}\) represents the inverse cosine function (arccosine). The objective is to determine the results using these inverse functions.

Step 2: Evaluating \\(\tan^{-1}(\tan(2.1))\\)

The function \(\tan(x)\) is periodic with a period of \(\pi\) (or approximately 3.14159). The range of \(\tan^{-1}(x)\) is \((-\pi/2)\) to \((\pi/2)\). Since 2.1 is within this interval, \(\tan^{-1}(\tan(2.1))\) will return 2.1 directly.

Step 3: Evaluating \\(\cos^{-1}(\cos(2.1))\\)

The function \(\cos(x)\) has a range between -1 and 1, and \(\cos^{-1}(x)\) gives an angle between \(0\) and \(\pi\) (or 0 to approximately 3.14159). Since 2.1 is within this range, and because the inverse function chooses the principal value, \(\cos^{-1}(\cos(2.1))\) returns 2.1 directly.

Step 4: Summary of Results

Given the domain restrictions for \(\tan^{-1}\) and \(\cos^{-1}\), both expressions, \(\tan^{-1}(\tan(2.1))\) and \(\cos^{-1}(\cos(2.1))\), return the value 2.1. This is because 2.1 lies within the range where the inverse functions return the principal values.

Key Concepts

Arctangent

The arctangent, often written as \( \tan^{-1}(x) \), is the inverse of the tangent function. It provides the angle whose tangent is a given number. One of the key aspects of the arctangent is its range, which is restricted to \((-\pi/2, \pi/2)\). This limited domain ensures that each input has a unique output, making the function well-defined and single-valued.

When evaluating expressions like \( \tan^{-1}(\tan(2.1)) \), the resulting value is straightforward if the angle \(2.1\) is within this range. Since \(2.1\) lies within \((-\pi/2, \pi/2)\), \( \tan^{-1}(\tan(2.1)) = 2.1 \). This results from the function 'undoing' itself perfectly within its principal branch.

• The arctangent provides principal values only within its range.
• Helps determine angles based on given tangent values.
• The periodic nature of the tangent function is accounted for by this specific inverse operation.

Arccosine

The arccosine, denoted as \( \cos^{-1}(x) \), is the inverse of the cosine function. This function retrieves the angle whose cosine is the number \(x\). The range of arccosine is from \(0\) to \(\pi\), aligning with the idea of principal values, where each angle within this interval corresponds to a unique cosine result.

In exercises like \( \cos^{-1}(\cos(2.1)) \), if \(2.1\) fits within the arccosine range, it will return \(2.1\) as the result. The value \(2.1\) being an angle in radians falls within \(0\) to \(\pi\), thus \( \cos^{-1}(\cos(2.1)) = 2.1 \). It's pivotal here that this particular angle lies within the principal interval.

• The arccosine ensures unique angle results for each cosine value.
• Conforms to a range from \(0\) to \(\pi\), a half-circle.
• Relies on the principal values to maintain the inverse function's integrity.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. Both tangent and cosine have distinct periodic behaviors:

• The tangent function, \( \tan(x) \), repeats every \(\pi\), or roughly every 3.14159 radians.
• Meanwhile, the cosine function, \( \cos(x) \), repeats every \(2\pi\).

This periodic nature means that for these functions, any angle outside one cycle can often look equivalent to an angle within the cycle range due to their symmetry and repetition.

However, when using inverse functions, we restrict the periodicity to principal values to ensure each inverse operation yields a single unique value and maintains the function's integrity. Thus,

• In \( \tan^{-1}(\tan(x)) \), the function will only return values in \((-\pi/2, \pi/2)\).
• For \( \cos^{-1}(\cos(x)) \), outputs range from \(0\) to \(\pi\).

Principal Values

Principal values are specific outputs of inverse trigonometric functions that lie within their restricted ranges. These values are crucial as they ensure that each function is inversely unique and single-valued.

For example, \( \tan^{-1} \) outputs values between \((-\pi/2)\) and \((\pi/2)\), while \( \cos^{-1} \) outputs angles between \(0\) and \(\pi\). With these restrictions, calculating things like \( \tan^{-1}(\tan(2.1)) \) and \( \cos^{-1}(\cos(2.1)) \), allows us to retrieve principal values directly if \(2.1\) is within these ranges.

• Ensures the function returns a single value by restricting the range.
• Aids in resolving ambiguities inherent in periodic functions.
• Maintains the inverse functional nature by providing unique outcomes.