Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the graph of \(y=\pi / 2-\arcsin x\). Make a conjecture. Prove it.

Step-by-Step Solution

Answer

The graph y=\(\pi/2-\arcsin x\) reflects \(\arcsin x\) across \(y=\pi/2\).

Step 1: Identify the Range and Domain

The function \(y = \pi/2 - \arcsin x\) involves the inverse sine function, \(\arcsin x\), which is defined only for \(x\) in the interval \([-1, 1]\). Therefore, the domain of the function is \([-1, 1]\). The range of \(\arcsin x\) is \([-\pi/2, \pi/2]\), so the range of \(y = \pi/2 - \arcsin x\) will be \([0, \pi]\).

Step 2: Analyze the Transformation of the Graph

The function \(y = \arcsin x\) is flipped vertically and shifted by \(\pi/2\) because of the expression \(\pi/2 - \arcsin x\). Therefore, it will invert the values of the \(\arcsin x\) graph and shift them upward by \(\pi/2\) units.

Step 3: Sketch the Graph

To draw the graph, start by considering key points from \(y = \arcsin x\). For \(x = -1\), \(\arcsin(-1) = -\pi/2\), making \(y = \pi/2 - (-\pi/2) = \pi\). For \(x = 0\), \(\arcsin(0) = 0\), making \(y = \pi/2 - 0 = \pi/2\). For \(x = 1\), \(\arcsin(1) = \pi/2\), making \(y = \pi/2 - \pi/2 = 0\). Plot these points and draw a smooth curve between them to complete the graph. The graph is decreasing on \([-1, 1]\).

Step 4: Make a Conjecture

Based on the graph, one can conjecture that \(y = \pi/2 - \arcsin x\) is the reflection of \(\arcsin x\) across the line \(y = \pi/2\).

Step 5: Prove the Conjecture

To prove the conjecture, consider that if \(y = \pi/2 - \arcsin x\), then rearranging gives \(\arcsin x = \pi/2 - y\). Plugging this into the equation yields \(x = \sin(\pi/2 - y)\). Using the trigonometric identity, \(\sin(\pi/2 - y) = \cos(y)\), we can express \(x = \cos(y)\). Therefore, the function indeed transforms points as hypothesized, confirming the reflection about the line \(y = \pi/2\).

Key Concepts

Graph Analysis

The graph of the function \(y = \frac{\pi}{2} - \arcsin x\) is intriguing, as it involves an inverse trigonometric function. When analyzing this graph, we start by identifying crucial points derived from the function \(y = \arcsin x\). \(y = \arcsin x\) itself is defined for \(x\) in the range -1 to 1, moving from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
\(y = \frac{\pi}{2} - \arcsin x\) repositions these values, reflecting them vertically about the line \(y=\frac{\pi}{2}\). This means the graph starts at \(\pi\) when \(x=-1\), descends to \(\frac{\pi}{2}\) at \(x=0\), and reaches 0 at \(x=1\). The graph's smooth curve signifies a consistent rate of change, gradually decreasing across its domain.

Function Transformation

The function \(y = \frac{\pi}{2} - \arcsin x\) involves both reflection and vertical translation. The expression \(\frac{\pi}{2} - \arcsin x\) can be broken down into two operations:

• A vertical shift: The constant \(\frac{\pi}{2}\) shifts the graph of \(\arcsin x\) upwards by \(\frac{\pi}{2}\) units.
• A vertical reflection: By subtracting \(\arcsin x\) from \(\frac{\pi}{2}\), the graph experiences a reflection across the line \(y = \frac{\pi}{2}\).

This transformation flips the outputs of \(\arcsin x\), giving us a new graph that is inverted across this central line, essentially reversing the function's behavior while perfectly mirroring it on its new axis.

Domain and Range

Understanding the domain and range of a function is crucial in graphing inverse trigonometric functions like \(y = \frac{\pi}{2} - \arcsin x\). In this case, the domain is determined by the arcsine function, \(-1 \leq x \leq 1\).
The range for \(\arcsin x\) being \([-\frac{\pi}{2}, \frac{\pi}{2}]\), results in the transformed range for \(y = \frac{\pi}{2} - \arcsin x\) being \([0, \pi]\).

• The lowest output of the function when \(x = 1\) is 0.
• Its highest output when \(x = -1\) is \(\pi\).

This transformation of range reflects how each input's effect on \(\arcsin x\) now translates to a new output after the transformation, highlighting the role of the vertical shift and reflection.

Trigonometric Identities

Trigonometric identities assist dramatically in interpreting transformations within inverse trigonometric functions. In \(y = \frac{\pi}{2} - \arcsin x\), our focus shifts to the identity \(\sin(\frac{\pi}{2} - y) = \cos(y)\). This identity confirms our conjecture about the reflection by rewriting the equation for transformation.
Starting with \(y = \frac{\pi}{2} - \arcsin x\), we rearrange to \(\arcsin x = \frac{\pi}{2} - y\). Substituting this back to the definition of sine, we find \(x = \cos(y)\), illustrating that the sine and cosine identity achieves our desired mirroring effect.

• This confirms \(x\)'s output equivalent to cosine's transformation from sine's inverse.
• It offers profound insight into how trigonometric functions complement each other through symmetric properties.

Understanding these identities helps to bridge the conceptual gap between raw computation and graphical or real-world interpretation of trigonometric functions.