Which trigonometric functions have a domain of $[-1,1]$?

• A. The sine and cosine functions ($\sin$ and $\cos$)
• B. The tangent and cotangent functions ($\tan$ and $\cot$)
• C. The secant and cosecant functions ($\sec$ and $\csc$)
• D. The inverse sine and inverse cosine functions ($\arcsin$ and $\arccos$)

Answer: (D)

Verified step by step guidance

1. Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. Identify the domains of the basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions typically have domains of all real numbers or all real numbers except where they are undefined (like tan and cot at certain points).
3. Understand that the inverse trigonometric functions, such as arcsin (inverse sine) and arccos (inverse cosine), have restricted domains because they reverse the original functions and must be one-to-one.
4. Recall that the domain of the inverse sine function, \(\arcsin(x)\), and the inverse cosine function, \(\arccos(x)\), is limited to the interval \([-1, 1]\) because sine and cosine values only range between -1 and 1.
5. Therefore, the trigonometric functions with domain \([-1, 1]\) are the inverse sine and inverse cosine functions, \(\arcsin\) and \(\arccos\).