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Chapter 5: Problem 81

Explain why the domains of the trigonometric functions are restricted when finding the inverse trigonometric functions.

Short Answer

Expert verified
The domains of the trigonometric functions are restricted because they are periodic and not one-to-one over their entire domains. To find an inverse that is a function, the domain of the original function must be restricted to an interval where it is one-to-one.

Step by step solution

01

Understand function and its inverse

A function is an operation that takes an input, or a set of inputs, and maps it to an output. The inverse of a function is the operation that reverses the effect of the function. That is, if a function takes an input 'a' and maps it to an output 'b', then the inverse function takes an input 'b' and maps it back to output 'a'.
02

Understand properties of trigonometric functions

All of the six standard trigonometric functions - sine, cosine, tangent, cosecant, secant, and cotangent - are periodic. This means that they repeat their values in regular intervals or 'periods'. For example, the sine and cosine functions repeat every \(2\pi\) radians, or 360 degrees.
03

Understand implications for inverses of trigonometric functions

Because the trigonometric functions are periodic, they do not pass the 'horizontal line test' for all real numbers, meaning that they are not one-to-one functions over their entire domains. To define an inverse that is a function, we must restrict the domain of the original function to a specific interval where it is one-to-one. For example, the sine function is restricted to the interval \([- \pi/2, \pi/2]\), and the cosine function is restricted to the interval \([0, \pi]\). These restricted functions are one-to-one, and their inverses are functions.

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