Problem 46 - Algebra and Trigonometry

Question

\(46-47\) . Graphing Inverse Trigonometric Functions (a) Graph the function and make a conjecture, and (b) prove that your conjecture is true. $$ y=\sin ^{-1} x+\cos ^{-1} x $$

Step-by-Step Solution

Answer

The graph and identity show \( y = \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \) for \( x \) in \([-1,1]\).

Step 1: Understand the Function

The function given is \( y = \sin^{-1} x + \cos^{-1} x \). Our goal is to graph this function and then make a conjecture about it.

Step 2: Graphing the Function

Let's consider the values of \( x \) from \(-1\) to \(1\) because \(\sin^{-1} x\) and \(\cos^{-1} x\) are only defined in this range. By calculating specific points, you will notice that for any value of \( x \), \( y \) will equal \( \frac{\pi}{2} \). This suggests a conjecture that \( y = \frac{\pi}{2} \) for all \( x \) in the domain.

Step 3: Conjecture

Based on our graph, the conjecture is that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \) for all \( x \) in the interval \([-1, 1]\).

Step 4: Prove the Conjecture

Recall the identity of trigonometric functions: \( \sin(\theta) + \cos(\theta) = 1 \) when \( \theta \) is an angle in a right triangle. For angles, \( \theta = \sin^{-1}(x) \) and \( \phi = \cos^{-1}(x) \), we have \( \theta + \phi = \frac{\pi}{2} \). Thus, \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), proving our conjecture.

Key Concepts

Graphing Functions

Graphing functions begins by understanding their definitions and domains. For the exercise at hand, you're dealing with inverse trigonometric functions such as \( \sin^{-1} x \) and \( \cos^{-1} x \). These functions are helpful because they allow you to find angles given trigonometric values.

When graphing \( y = \sin^{-1} x + \cos^{-1} x \), the first step is to identify where these functions are defined. Both the sine and cosine inverse functions are confined to the interval \([-1, 1]\). This means that your graph will also be confined to these x-values.

As you plot points for \( x \), you'll observe that the y-values are constant. Specifically, you find that \( y = \frac{\pi}{2} \) for all points in the interval. This forms a horizontal line on the graph, showcasing a unique behavior of inverse trigonometric functions when their results are summed.

Trigonometric Identities

Trigonometric identities are relationships between trigonometric functions that hold true for all possible values of the variables. In this context, one of the key identities is the sum of the inverse sine and cosine functions: \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \).

To understand this identity, consider a right triangle. If one angle is \( \theta \), then the other non-right angle must be \( \frac{\pi}{2} - \theta \). So, the identities \( \sin(\theta) = x \) and \( \cos(\frac{\pi}{2} - \theta) = x \) translate into the inverse functions summing to \( \frac{\pi}{2} \).

Such identities are incredibly powerful tools when working with trigonometric functions and are the bedrock for many algebraic manipulations in trigonometry.

Domain and Range

Understanding the domain and range of a function is essential in graphing and interpreting its behavior. For the functions \( \sin^{-1} x \) and \( \cos^{-1} x \), the domain is limited to \([-1, 1]\). This is because these functions are defined only when \( x \) falls within this interval.

The range of \( \sin^{-1} x \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), while for \( \cos^{-1} x \), it is \([0, \pi]\). However, when combined to form the new function \( \sin^{-1} x + \cos^{-1} x \), the resultant range remains solely as \( \frac{\pi}{2} \), which is a constant.

The specific domain and range of these inverse trigonometric functions guide you in setting up and interpreting their graphs correctly, ensuring you stay within the bounds of their possible values.