Problem 228 - Calculus Volume 1

Question

[T] Use a calculator to evaluate \(\sin \left(\sin ^{-1}(-2)\right)\) and \(\tan \left(\tan ^{-1}(-2)\right) .\) Explain the results of each.

Step-by-Step Solution

Answer

\(\sin(\sin^{-1}(-2))\) is undefined; \(\tan(\tan^{-1}(-2)) = -2\).

Step 1: Understanding Inverse Functions

The inverse sine function, denoted as \( \sin^{-1}(x) \), only has defined values for \( x \) between \(-1\) and \(1\). Therefore, \( \sin^{-1}(-2) \) is not possible because \(-2\) lies outside this range. Similarly, the inverse tangent function, denoted as \( \tan^{-1}(x) \), can take any real number as input.

Step 2: Evaluate \(\sin(\sin^{-1}(-2))\) with a Calculator

Since \(\sin^{-1}(-2)\) is undefined, \(\sin(\sin^{-1}(-2))\) cannot be evaluated using a calculator or otherwise. The domain restrictions of the inverse sine function prevent it from having a real output for the input \(-2\).

Step 3: Evaluate \(\tan(\tan^{-1}(-2))\) with a Calculator

The function \(\tan(\tan^{-1}(x))\) will return \(x\) for any real number, as \(\tan^{-1}(x)\) outputs the angle whose tangent is \(x\). Using a calculator, \(\tan(\tan^{-1}(-2)) = -2\).

Step 4: Interpreting the Results

For \(\tan(\tan^{-1}(-2))\), the result is \(-2\) because the function \(\tan\) and its inverse \(\tan^{-1}\) perfectly revert the transformation back to the original number when the input is valid for the inverse function's domain and range. For \(\sin(\sin^{-1}(-2))\), the result is undefined.

Key Concepts

Understanding the Inverse Sine Function

The inverse sine function, represented as \( \sin^{-1}(x) \), is an important tool in trigonometry for finding angles when a sine value is known. However, it is critical to understand the restrictions that apply to this function. The domain of the inverse sine function is restricted. It can only handle values within the range of \(-1\) to \(1\). This limitation arises because these are the maximum and minimum values that a sine function can naturally produce. Therefore, if you try to compute \( \sin^{-1}(-2) \), it's not possible. This is because \(-2\) does not fall within the valid input range. Such an operation results in 'undefined'. This restriction keeps the output within the bounds of real numbers, which are typically angles measured in radians or degrees.

Exploring the Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \), is used to ascertain the angle whose tangent is a given number. Interestingly, the inverse tangent function allows for any real number as input. This characteristic sets it apart from the inverse sine function. So, when you compute \( \tan^{-1}(-2) \), the output is an angle. This angle is such that the tangent of it will return \(-2\). This is an unrestricted domain and thus makes \( \tan^{-1} \) versatile for calculations involving any real number. The evaluation of \( \tan(\tan^{-1}(-2)) \) simply returns \(-2\). The inverse and original function cancel each other out effectively within their operating domain, much like a combination lock returning to the start after inputting the correct code.

Grasping Domain Restrictions

Domain restrictions are central to understanding how inverse trigonometric functions operate. These restrictions define what input values are acceptable for a function, influencing whether or not you can achieve a tangible output.

• Inverse Sine Function: restricted to inputs between \(-1\) and \(1\).
• Inverse Tangent Function: can handle all real numbers.

When dealing with these functions, remember that trying to input values outside the permissible domain leads to undefined results. This is a failsafe in mathematical operations, ensuring we only work with numbers that are meaningful in their respective contexts. Appreciating these domain limits is key when evaluating complex trigonometric formulas.