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Chapter 4: Problem 99

Explain why, without restrictions, no trigonometric function has an inverse function.

Short Answer

Expert verified
The trigonometric functions do not have an inverse without restrictions due to their periodic nature, repeating the values over their periods. This makes them non-bijective, violating the definition of an inverse function. However, by restricting the domain, they can become bijective, allowing for the existence of an inverse.

Step by step solution

01

Understanding Trigonometric Functions

Express trigonometric functions like sine, cosine, or tangent and sketch them. For example, for the sine function, it varies between -1 and 1, and repeats itself every \(2\pi\) radians. Hence, for every value in the range of the function, there are infinitely many corresponding values in the domain. This means the function is not one-to-one.
02

Understanding Inverse Functions

An inverse function is a function that reverses the mapping of the original function, essentially reflecting it along the line \(y=x\). However, for a function to have an inverse, it must be bijective (a one-to-one and onto function). This means each value in the range corresponds to one and only one value in the domain.
03

Conclusion

Trigonometric functions are not one-to-one functions due to their periodic nature, which means they lack an inverse in general. However, if we restrict the domain of these functions to a certain interval where they are one-to-one, an inverse can be defined. For example, if we restrict the domain of the sine function to \([- \pi /2, \pi /2]\), then it's possible to define an inverse.

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