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Chapter 4: Problem 81

Simplify \(\sin ^{-1}(\sin x)\) if \(-\pi / 2 \leq x \leq \pi / 2\).

Short Answer

Expert verified
(x)=x\, \text{for}\, -\pi/2 \leq x \leq \pi/2.

Step by step solution

01

Understand the Function

The function \(\sin^{-1}(\sin x)\) is the inverse sine function applied to the sine of \(x\). The inverse sine function, \(\sin^{-1}\), outputs angles in the range \(-\pi/2, \pi/2\) inclusive.
02

Determine the Domain of x

Given the interval \(-\pi/2 \leq x \leq \pi/2\),\, \(x\) itself is already in the output range of the inverse sine function. Thus, when calculating \(\sin^{-1}(\sin x)\), \(x\) is unaffected.
03

Simplification Conclusion

Since \(x\) is within the interval \([-\pi/2, \pi/2]\), the expression \(\sin^{-1}(\sin x)\) simplifies directly to \(x\). This is because the inverse operation applied to a value within its range returns the original value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplification of Expressions
Simplification is about reducing expressions to their simplest form. In the case of \( \sin^{-1}(\sin x) \), simplifying means understanding how inverse and trigonometric functions interact.

To simplify \( \sin^{-1}(\sin x) \), it's essential to recognize the properties of the sine and inverse sine functions. The inverse sine function, \( \sin^{-1} \), reverses the effect of the sine function, returning the original angle if that angle is within the valid range.

For \( x \) in the interval \([-\pi/2, \pi/2]\), the sine of \( x \) is itself within the range of values that the inverse sine function can handle directly, \([-1, 1]\).
  • Therefore, if \( x \) falls within this range, the entire expression simplifies to just \( x \).
  • This is a powerful result, as it's based on properly applying the properties of these trigonometric functions.
Domain and Range
Domain and range are fundamental concepts in understanding functions. The domain is the set of all possible input values, while the range is the set of all possible output values.

The original exercise deals with \( \sin^{-1}(\sin x) \) where \(-\pi/2 \leq x \leq \pi/2\). In this case, the domain of \( x \) conveniently matches the range of the inverse sine function.
  • The inverse sine function's range is \([-\pi/2, \pi/2]\), which aligns perfectly with our given interval.
  • When an expression's domain matches the range of the inverse function, simplification becomes straightforward because the inverse function will output the original input.
Knowing the domain and range helps predict how functions will behave, ensuring that operations like simplification are handled correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. They are crucial for simplifying and solving trigonometric equations.

In this context, understanding the identity \( \sin^{-1}(\sin x) = x \) for \( x \) in the interval \([-\pi/2, \pi/2]\) highlights the importance of these identities.
  • These identities rely on the fundamental properties of trigonometric functions, such as periodicity and symmetry.
  • The identity works because within the specified range, the sine and its inverse perfectly counteract each other.
These identities are powerful tools in trigonometry, providing neat and simple pathways to solve complex problems.

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Most popular questions from this chapter

Identify the period for each of the following. Do not sketch the graph. $$ y=\cos \frac{\pi}{2} x $$

Find the period of \(y=3 \cot \left(\frac{x}{4}\right)\). a. \(\pi\) b. \(4 \pi\) c. \(8 \pi\) d. \(\frac{\pi}{4}\)

The problems that follow review material we covered in Section 3.2. Reviewing these problems will help you with the next section. Evaluate each of the following if \(x\) is \(\pi / 2\) and \(y\) is \(\pi / 6\). $$ \sin \left(y-\frac{\pi}{2}\right) $$

Graph each of the following over the given interval. Label the axes so that the amplitude and period are easy to read. $$ y=3 \cos \pi x,-2 \leq x \leq 4 $$

Identify the amplitude for each of the following. Do not sketch the graph. $$ y=-\frac{1}{4} \cos x $$
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