Problem 2 - Algebra 2 and Trigonometry

Question

Sammy said that if a negative value is entered for \(\sin ^{-1}, \cos ^{-1},\) or \(\tan ^{-1}\) , the calculator will return a negative value for the measure of the angle. Do you agree with Sammy? Explain why or why not.

Step-by-Step Solution

Answer

Sammy is partially correct; \( \sin^{-1} \) and \( \tan^{-1} \) can return negative values, but \( \cos^{-1} \) cannot.

Step 1: Understand Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the process of standard trigonometric functions. For instance, if \( y = \sin(x) \), then \( x = \sin^{-1}(y) \). These functions are used to find the angle corresponding to a given trigonometric value.

Step 2: Recall the Range of Inverse Functions

The inverse trigonometric functions have specific ranges due to their periodic nature and to make them functions. The range for \( \sin^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), for \( \cos^{-1}(x) \) is \([0, \pi]\), and for \( \tan^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 3: Analyze Negative Inputs

When negative values are entered into \( \sin^{-1} \) or \( \tan^{-1} \), the calculator can indeed return negative angle values since their ranges include negative angles. However, \( \cos^{-1} \) does not return negative angles for any value because its range does not include negative numbers.

Step 4: Conclusion Based on Range Analysis

Based on the analysis of the ranges, \( \sin^{-1} \) and \( \tan^{-1} \) can return negative values. However, \( \cos^{-1} \) will always output a non-negative angle.

Key Concepts

Range of Inverse Functions

Inverse trigonometric functions are essential in calculus and trigonometry because they provide the angle corresponding to a given trigonometric value. Each of these functions has a distinct range due to the periodic nature of trigonometric functions. Their ranges ensure that each inverse function outputs a single value for a specific input, which is necessary for them to be true functions and not just relations.

• The range of \( \sin^{-1}(x) \), also known as arcsin, is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means it will return angles in the first and fourth quadrants, which include both positive and negative values.
• The range of \( \cos^{-1}(x) \), or arccos, is \([0, \pi]\). This function will only return angles within these bounds, covering the first and second quadrants, hence only non-negative outputs.
• The range of \( \tan^{-1}(x) \), or arctan, is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), similar to arcsin, allowing for both negative and positive angle outputs.

Due to these specified ranges, you can determine the specific measure of an angle from its trigonometric value.

Negative Inputs in Trigonometric Functions

When plugging negative values into trigonometric functions, the type of inverse function you are using is crucial in determining the nature of the returned angle. Not all inverse functions react similarly to negative inputs. For \( \sin^{-1}(x) \) and \( \tan^{-1}(x) \), having ranges that extend into negative values, the calculator can return negative angles when negative trigonometric values are used as inputs. These functions are designed to provide outputs that mirror the inputs' signs within their respective ranges. However, for \( \cos^{-1}(x) \), no matter if you input a negative or a positive value, the output will always be non-negative. This results from the defined range \([0, \pi]\), which restricts the outputs to positive angles only. Thus, when you input negative values, the behavior of the inverse trigonometric function is directly influenced by its range, leading to different possible outcomes.

Function Range in Trigonometry

Understanding function ranges is paramount in mastering how trigonometric functions and their inverses work. The range signifies the possible outputs for a given function. For inverse trigonometric functions:

• \( \sin(x) \) outputs values between \(-1\) and \(1\), making its inverse, \( \sin^{-1}(x) \), responsible for angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• \( \cos(x) \) also has outputs ranging from \(-1\) to \(1\), with \( \cos^{-1}(x) \) returning angles from \(0\) to \(\pi\). This indicates no negative angle output.
• \( \tan(x) \) spans all real numbers, necessitating \( \tan^{-1}(x) \) having a range of \([-\frac{\pi}{2}, \frac{\pi}{2})\).

Grasping these ranges is key in determining the behavior of inverse trigonometric functions for any given input, especially when negative values are involved. Being aware of these limitations helps prevent errors in calculations and in the interpretation of trigonometric results.