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draw the graph:\sin \left(\sin ^{-1}x\right)
Feb 12, 2024
draw the graph:\sin \left(\sin ^{-1}x\right)
Generated Graph
Solution by Steps
step 1
To draw the graph of the function y=sin⁡(sin⁡−1(x)) y = \sin(\sin^{-1}(x)) , we recognize that sin⁡−1(x)\sin^{-1}(x) is the inverse function of sin⁡(x)\sin(x)
step 2
The function sin⁡(sin⁡−1(x)) \sin(\sin^{-1}(x)) simplifies to y=x y = x for all x x in the domain of sin⁡−1(x)\sin^{-1}(x), which is [−1,1][-1, 1]
step 3
The graph of y=x y = x is a straight line passing through the origin with a slope of 1
step 4
Plot the straight line for x x values between -1 and 1 to represent the graph of y=sin⁡(sin⁡−1(x)) y = \sin(\sin^{-1}(x))
Answer
The graph is a straight line passing through the origin with a slope of 1, within the interval [−1,1][-1, 1].
Key Concept
Inverse Functions and Their Composition
Explanation
The composition of a function and its inverse results in the identity function over the domain of the inverse function. In this case, sin⁡(sin⁡−1(x))=x \sin(\sin^{-1}(x)) = x for x x in [−1,1][-1, 1], which is graphed as a straight line.
draw the graph:\sin ^{-1}\left(\sin x\right)
Generated Graph
Solution by Steps
step 1
To draw the graph of y=sin⁡−1(sin⁡x) y = \sin^{-1}(\sin x) , we need to understand the behavior of the sine function and its inverse
step 2
The sine function, sin⁡x \sin x , oscillates between -1 and 1 for all x
step 3
The inverse sine function, sin⁡−1x \sin^{-1}x , is defined for the domain [−1,1] [-1, 1] and has a range of [−π2,π2] [-\frac{\pi}{2}, \frac{\pi}{2}]
step 4
When we compose sin⁡−1 \sin^{-1} with sin⁡x \sin x , the output is x whenever x is in the range [−π2,π2] [-\frac{\pi}{2}, \frac{\pi}{2}] . Outside this interval, the graph will repeat periodically due to the periodic nature of the sine function
step 5
The graph of y=sin⁡−1(sin⁡x) y = \sin^{-1}(\sin x) will be a series of line segments with a slope of 1, where each segment spans an interval of length π \pi and is centered at multiples of π \pi
Answer
The graph of y=sin⁡−1(sin⁡x) y = \sin^{-1}(\sin x) consists of line segments with a slope of 1, repeating every π \pi units along the x-axis.
Key Concept
Composition of a function and its inverse
Explanation
The graph of y=sin⁡−1(sin⁡x) y = \sin^{-1}(\sin x) shows the output of x whenever x is within the principal range of [−π2,π2] [-\frac{\pi}{2}, \frac{\pi}{2}] for the inverse sine function. Outside this range, the graph will exhibit a sawtooth pattern due to the periodic nature of the sine function and the restricted domain of its inverse.
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