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Chapter 5: Problem 100

Determine whether the statement is true or false. Justify your answer. If you correctly solve a trigonometric equation to the statement \(\sin x=3.4,\) then you can finish solving the equation by using an inverse function.

Short Answer

Expert verified
The statement is false, as the equation \(\sin x=3.4\) does not have a valid solution, and therefore cannot be solved using an inverse function.

Step by step solution

01

Analyze the trigonometric statement

Given the equation \(\sin x=3.4,\) we know that the sine function outputs values between -1 and 1 for any real number \(x\) in its domain. Therefore, \(\sin x\) cannot be equal to 3.4, as 3.4 lies outside the range of -1 to 1.
02

Equation validity verification

Because the equation \(\sin x=3.4\) is not valid, it means that it cannot be solved through any method, including using an inverse function. The inverse function, \(\sin^{-1}(y)\), is defined only for values of \(y\) in the interval [-1,1], which does not include 3.4.
03

Conclusion

Therefore, the given statement is false. You cannot correctly solve this trigonometric equation to the statement \(\sin x=3.4,\) and then finish solving the equation by using an inverse function, because \(\sin x\) can never equal 3.4. Using an inverse function is not possible here as the value 3.4 lies outside the valid range for the sine function.

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