Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the graph of \(y=\sin (\arcsin x)\) on \([-1,1]\). Then draw the graph of \(y=\arcsin (\sin x)\) on \([-2 \pi, 2 \pi]\). Explain the differences that you observe.

Step-by-Step Solution

Answer

The graph of \(y = \sin(\arcsin x)\) is a straight line, while \(y = \arcsin(\sin x)\) is piecewise linear with breaks at multiples of \(\pi\).

Step 1: Understand the function y = sin(arcsin(x))

For the function \(y = \sin(\arcsin x)\), we need to understand the composition. The \(\arcsin(x)\) function gives the angle whose sine is \(x\). Since the range of \(\arcsin(x)\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the \(\sin(\arcsin(x))\) essentially returns \(x\) for all \(x\) in the domain \([-1, 1]\). Thus, the graph of \(y = \sin(\arcsin x)\) is a straight line \(y = x\) from \(-1\) to \(1\).

Step 2: Graph y = sin(arcsin(x))

The graph of \(y = \sin(\arcsin x)\) on the interval \([-1, 1]\) is simply a diagonal line with a slope of 1 passing through the origin. This is because \(\sin(\arcsin x) = x\). The endpoints are \((-1, -1)\) and \((1, 1)\).

Step 3: Understand the function y = arcsin(sin(x))

For the function \(y = \arcsin(\sin x)\), you need to consider that the composition's result depends on the range of \(\arcsin\), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Within this interval, \(\arcsin(\sin x) = x\). Outside this range, \(\arcsin(\sin x)\) returns the equivalent angle within \([-\frac{\pi}{2}, \frac{\pi}{2}]\). For instance, \(\arcsin(\sin \pi) = 0\).

Step 4: Graph y = arcsin(sin(x))

To sketch \(y = \arcsin(\sin x)\) on the interval \([-2\pi, 2\pi]\), note that this is a piecewise function due to the periodic nature of sine and the restricted range of arcsin. From \(-2\pi\) to \(-\frac{3\pi}{2}\), and from \(\frac{3\pi}{2}\) to \(2\pi\), the graph will be a straight line. From \(-\frac{3\pi}{2}\) to \(-\pi\) and from \(\pi\) to \(\frac{3\pi}{2}\), it will follow a negative linear line. Between \(-\pi\) to \(\pi\), it will align with \(y = x\). The resulting pattern visually exhibits symmetry and periodicity.

Step 5: Compare the Graphs

The graph of \(y = \sin(\arcsin x)\) is a simple, straight diagonal line from \(-1\) to \(1\). In contrast, the graph of \(y = \arcsin(\sin x)\) becomes piecewise linear over \([-2\pi, 2\pi]\), due to the limited range of arcsin. Each segment is linear with jumps at multiples of \(\pi\), keeping the range within \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, \(\sin(\arcsin x)\) undoes the arcsine perfectly, while \(\arcsin(\sin x)\) remains confined by its principal range.

Key Concepts

Domain and Range

The domain of a function consists of all the possible input values that the function can accept, while the range encompasses all the output values the function can produce. For trigonometric functions like \( ext{sin}\) and \( ext{arcsin}\), understanding these aspects is crucial.
For the function \(y = ext{sin}( ext{arcsin}(x))\), the domain is \([-1, 1]\). This is because the arcsine function, \( ext{arcsin}(x)\) or inverse sine, is only defined for inputs within this range. The range of \( ext{arcsin}(x)\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), but since we require the \( ext{sin}\) of \( ext{arcsin}(x)\), it effectively yields back the original value, making the range of the entire function also \([-1, 1]\).
This results in \(y = ext{sin}( ext{arcsin}(x))\) being a direct line\(y = x\) just like its domain.

• Domain of \(y = ext{sin}( ext{arcsin}(x))\): \([-1, 1]\)
• Range of \(y = ext{sin}( ext{arcsin}(x))\): \([-1, 1]\)

For \(y = ext{arcsin}( ext{sin}(x))\), the situation is different. Its domain extends over a larger interval \([-2\pi, 2\pi]\). However, its range remains constrained by the principal range of \( ext{arcsin}\), namely \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This range restriction causes the graph to adjust itself into a piecewise manner, adapting parts of \( ext{sin}(x)\) into this confined range. This complexity is key to grasping this function's graphical behavior.

• Domain of \(y = ext{arcsin}( ext{sin}(x))\): \([-2\pi, 2\pi]\)
• Range of \(y = ext{arcsin}( ext{sin}(x))\): \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

Piecewise Functions

Piecewise functions are defined by different expressions depending on the subdomain. For \(y = ext{arcsin}( ext{sin}(x))\), the function has to 'break' into segments due to the periodic nature of sine and the limited range of \( ext{arcsin}\).
This means we must treat the function as distinct pieces when graphing it over \([-2\pi, 2\pi]\). Within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the graph aligns perfectly with \(y = x\). However, as \(x\) moves out of this range, the \( ext{arcsin}\) function re-adjusts to output values within its principal range, yielding a repeat pattern in linear form. This is why the graph is not a smooth curve but discrete linear parts forming a zig-zag or sawtooth pattern.
By using a piecewise approach for such trigonometric functions, we can simplify complex behavior into more manageable parts:

• From \([-2\pi, -\frac{3\pi}{2}]\) and \([\frac{3\pi}{2}, 2\pi]\), \(y = x + 2\pi\) and \(y = x - 2\pi\).
• From \([-\frac{3\pi}{2}, -\pi]\) and \([\pi, \frac{3\pi}{2}]\), \(y = x - \pi\) and \(y = x + \pi\).
• From \([-\pi, \pi]\), \(y = x\).

This careful partitioning is crucial for accurately understanding and graphing the intricate output of \( ext{arcsin}( ext{sin}(x))\).

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( ext{arcsin}\), are essential tools for finding angles based on given trigonometric values. Their distinctive characteristic is their limited range, which maps any result into a specific interval to maintain function validity.
For example, the \( ext{arcsin}\) function will only output results between \([-\frac{\pi}{2}, \frac{\pi}{2}]\), regardless of the sine input, meaning whenever we involve \( ext{arcsin}\) in function composition like \(y = ext{arcsin}( ext{sin}(x))\), results are always mapped to this range. This limited range assures that every x only corresponds to a single value of \( ext{arcsin}\), making the function invertible and well-defined.
When looking at the graph for \(y = ext{arcsin}( ext{sin}(x))\), the 'reset' within the repeating pattern you observe is because of this restricted range. Values beyond the principal range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\) are brought back into it by selecting equivalent angles within this range, creating linear segments and jumps on the graph, specifically at multiples of \(\pi\).

• Consider \( ext{arcsin}(1)\) or \( ext{arcsin}(-1)\), both returning edge values: \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\) respectively.
• Any other sine value, say 0, returns 0 within \( ext{arcsin}\)'s range.

Thus, inverse trigonometric functions give us a new perspective on angles and their measures, crucial for handling periodic behavior correctly and graphically interpreting sine breathes within bounds. Always take note of their range, so your graphs clearly reflect their piece-wise behaviors.