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. Answers to Concepts Review: \(\quad\) 1. \([-\pi / 2, \pi / 2] ;\) arcsin 2\. \((-\pi / 2, \pi / 2)\); arctan 3\. \((-\infty, \infty) ;(-\pi / 2, \pi / 2)\) 4\. \(\sqrt{1-x^{2}}\)

Short answer

\(y = \sin(\arcsin x)\) is a line \(y=x\); \(y=\arcsin(\sin x)\) folds between \(-\frac{\pi}{2}, \frac{\pi}{2}\).

Step-by-step solution

Understand the function y=sin(arcsin x)

Consider the function \( y = \sin(\arcsin x) \). The expression \( \arcsin x \) represents the inverse sine function, which effectively reverses the sine function, meaning \( \arcsin(\sin(y)) = y \) for \( y \) in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Since \( \sin(\arcsin(x)) = x \) for \( x \in [-1, 1] \), the graph of \( y = \sin(\arcsin x) \) on \([-1,1]\) is a straight line \( y = x \).

Graph y=sin(arcsin x) on [-1,1]

Plot the graph of \( y = x \) for the interval \( [-1, 1] \). Since \( \sin(\arcsin(x)) = x \) over this domain, each input results directly in its own value as output without any alteration.

Understand the function y=arcsin(sin x)

The function \( y = \arcsin(\sin(x)) \) takes the sine of \( x \) and then applies the inverse sine. Since \( \arcsin \) is only properly defined over the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the output of \( y = \arcsin(\sin(x)) \) is limited to this interval. The graph will thus "bounce" between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) as \( x \) progresses outside this range.

Graph y=arcsin(sin x) on [-2π, 2π]

To plot \( y = \arcsin(\sin(x)) \) from \(-2\pi\) to \(2\pi\), observe behavior periodicity: for intervals shifted by periods of \([0, \pi), [\pi, 2\pi), [-\pi, 0)\), \( y \) mirrors itself as sin oscillates between 1 and -1. This results in a piecewise linear graph constrained to \([-\frac{\pi}{2}, \frac{\pi}{2}]\) despite input \( x \) extending beyond.

Observations of Differences

The graph of \( y = \sin(\arcsin(x)) \) is a straightforward line \( y = x \) within \([-1, 1]\) because \( \arcsin \) perfectly reverses \( \sin \) over that domain. In contrast, \( y = \arcsin(\sin(x)) \) is non-linear on the broader \([-2\pi, 2\pi]\) domain, folding its graph at the periodic boundaries to stay within \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This behavior results from how inverse trigonometric functions limit outputs.

Key concepts

Sine Function

The sine function, commonly denoted as \( \sin(x) \), is a fundamental trigonometric function. It defines the relationship between an angle in a right triangle and the ratio of the opposite side over the hypotenuse. In terms of the unit circle, the sine of an angle \( x \) corresponds to the y-coordinate of a point on the circle.
The sine function is periodic, meaning it repeats its values in regular intervals. For sine, this period is \( 2\pi \), so every \( 2\pi \) units, the function's output cycle repeats. Its range is limited between -1 and 1.
A visual representation or graph of the sine function shows its wave-like pattern, oscillating smoothly between its maximum and minimum values.

Arcsin Function

The arcsin function, denoted as \( \arcsin(x) \) or sometimes \( \sin^{-1}(x) \), is the inverse of the sine function. It takes a value between -1 and 1 and returns an angle ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This limited range ensures the function is bijective, allowing for a unique angle output for each input value.
Arcsin reverses the effect of sine but only within this restricted output range. When \( \sin(\theta) = x \), then \( \arcsin(x) = \theta \), where \( \theta \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
In graph terms, the \( y = \arcsin(x) \) curve is an increasing line that spans the input domain from -1 to 1 and maps it directly to angles in radians.

Graph Interpretation

Graph interpretation involves analyzing the visual portrayal of mathematical functions. When viewing graphs of \( y = \sin(\arcsin(x)) \) and \( y = \arcsin(\sin(x)) \), it's important to understand how input values are transformed.
For \( y = \sin(\arcsin(x)) \), the graph within the domain \([-1,1]\) is a straight line, simply showing \( y = x \). This outcome occurs because arcsin exactly "undoes" the sine within this domain.
Conversely, \( y = \arcsin(\sin(x)) \) is more complex on \([-2\pi, 2\pi]\). The graph "bounces" within \([-\frac{\pi}{2}, \frac{\pi}{2}]\), as arcsin maps the sine output back into this specific interval, creating a folded appearance on the graph.

Periodicity

Periodicity is a core characteristic of trigonometric functions, indicating that they repeat their values at regular intervals. For the sine function, every \( 2\pi \) interval marks a complete cycle, after which the values repeat identically.
This property is why, when examining \( y = \arcsin(\sin(x)) \), we observe a repetition in the output as \( x \) progresses from \(-2\pi \) to \(2\pi\). Nonetheless, because arcsin can only produce outputs within its restricted range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the repetition manifests as folding or bouncing at the boundaries of this range.
Understanding periodicity is crucial for analyzing the behavior of trigonometric graphs, especially those involving inverse functions.

Domain and Range Analysis

Determining the domain and range of functions like \( \sin \) and \( \arcsin \) is essential for understanding their behavior. The domain of a function is the set of all possible input values, while the range is the set of all potential outputs.
For \( \sin(x) \):

• Domain: \( (-\infty, \infty) \)
• Range: \([-1, 1]\)

For \( \arcsin(x)\):

• Domain: \([-1, 1]\)
• Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

When analyzing \( y = \arcsin(\sin(x)) \), we see an interesting effect due to the arcsin's range. Even though \( \sin(x) \) can produce any value between -1 and 1 for all \( x \), the arcsin limits the output to the narrower range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This follows the principle that the domain of one function becomes the range of its inverse, adjusted for periodic limitations.