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Chapter 6: Problem 56

Evaluate each expression exactly, if possible. If not possible, state why. $$\sin \left[\sin ^{-1}(1.1)\right]$$

Short Answer

Expert verified
The expression \( \sin \left[\sin^{-1}(1.1)\right] \) cannot be evaluated because \( \sin^{-1}(1.1) \) is undefined.

Step by step solution

01

Understanding Inverse Sine Function Range

The function \( \sin^{-1}(x) \) is the inverse of \( \sin(x) \) and is defined only for \( -1 \leq x \leq 1 \). This means that \( \sin^{-1}(1.1) \) is not defined, as 1.1 is outside of this range.
02

Interpretation of the Problem

Since \( \sin^{-1}(1.1) \) is not within the domain of the inverse sine function, the expression \( \sin \left[\sin^{-1}(1.1)\right] \) cannot be evaluated because the first part, \( \sin^{-1}(1.1) \), is not valid.
03

Conclusion

Due to the limitation of the domain of \( \sin^{-1}(x) \), the given expression \( \sin \left[\sin^{-1}(1.1)\right] \) is not possible to evaluate, as it involves an undefined operation.

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Key Concepts

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

Domain and Range of Inverse Functions
Understanding the domain and range of inverse functions is crucial when dealing with inverse trigonometric functions. The inverse sine function, denoted as \( \sin^{-1}(x) \), has a specific domain and range that you must adhere to. The domain is the set of all possible input values, while the range is the set of all possible output values. For \( \sin^{-1}(x) \), the domain is \(-1 \leq x \leq 1\). This means that any value of \( x \) outside this interval is not valid for the inverse sine function.
The output (or range) for \( \sin^{-1}(x) \) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). To evaluate expressions involving these functions correctly, it鈥檚 essential to ensure that your inputs fall within the defined domain. If they don't, as with \( \sin^{-1}(1.1) \), the function is undefined.
Function Evaluation: Understanding Limitations
Evaluating trigonometric functions often involves recognizing the limitations and constraints of these functions. Here, we tried to evaluate \( \sin \left[\sin^{-1}(1.1)\right] \). This process requires us first to evaluate the inner expression, \( \sin^{-1}(1.1) \), before applying the outer sine function.However, as established, the inner expression is not defined because 1.1 is beyond the domain of \( \sin^{-1}(x) \). Function evaluation hinges on whether or not the input values lie within acceptable bounds. Without valid input from the inner inverse function, the outer operation cannot proceed.
Thus, attempting to evaluate functions without ensuring their inputs are valid can lead to incorrect solutions, emphasizing the importance of understanding these limitations.
Sine Function: An Important Trigonometric Function
The sine function, denoted as \( \sin(x) \), is one of the foundational trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.The function is periodic, with a period of \( 2\pi \), meaning it repeats its values in intervals of \( 2\pi \). Its range is from -1 to 1, which describes all the possible values it can output. This characteristic is why its inverse function, \( \sin^{-1}(x) \), is restricted to the domain of \(-1 \leq x \leq 1\). Understanding how the sine function operates and interacts with its inverse is key when working with trigonometric expressions. Ensuring that inputs and outputs remain within range can prevent errors and lead to accurate function evaluations.

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Most popular questions from this chapter

Graphing calculators can be used to find approximate solutions to trigonometric equations. For the equation \(f(x)=g(x),\) let \(Y_{1}=f(x)\) and \(Y_{2}=g(x) .\) The \(x\) -values that correspond to points of intersections represent solutions. With a graphing utility, find all of the solutions to the equation \(\cos \theta=e^{\theta}\) for \(\theta \geq 0\).

Touch-tone keypads have the following simultaneous low and high frequencies. $$\begin{array}{|l|c|c|c|} \hline \text { Freauency } & 1209 \mathrm{Hz} & 1336 \mathrm{Hz} & 1477 \mathrm{Hz} \\ \hline 697 \mathrm{Ilz} & 1 & 2 & 3 \\ \hline 770 \mathrm{Hz} & 4 & 5 & 6 \\ \hline 852 \mathrm{Hz} & 7 & 8 & 9 \\ \hline 941 \mathrm{Hz} & ^{*} & 0 & \\# \\ \hline \end{array}$$ The signal given when a key is pressed is \(\sin \left(2 \pi f_{1} t\right)+\sin \left(2 \pi f_{2} t\right)\) where \(f_{1}\) is the low frequency and \(f_{2}\) is the high frequency. What is the mathematical function that models the sound of dialing \(3 ?\)

Determine whether each statement is true or false. The product of two cosine functions is a sum of two other cosine functions.

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$\cos x=2-\cos x, 0 \leq x \leq 2 \pi$$

Explain the mistake that is made. $$\text { Solve } \sqrt{3 \sin \theta-2}=-\sin \theta \text { on } 0 \leq \theta \leq 2 \pi$$ Solution: Square both sides. \(3 \sin \theta-2=\sin ^{2} \theta\). Gather all terms to one side. \(\sin ^{2} \theta-3 \sin \theta+2=0\). Factor. \(\quad(\sin \theta-2)(\sin \theta-1)=0\). Set each factor equal to zero. \(\sin \theta-2=0\) or \(\sin \theta-1=0\). Solve for \(\sin \theta, \quad \sin \theta=2 \quad\) or \(\quad \sin \theta=1\). Solve \(\sin \theta=2\) for \(\theta . \quad\) no solution. Solve \(\sin \theta=1\) for \(\theta . \quad \quad \theta=\frac{\pi}{2}\). This is incorrect. What mistake was made?
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