Question

True or False The only trigonometric functions whose domain is all real numbers are the sine and cosine functions.

Short answer

True

Step-by-step solution

- Understand the question

The question asks if the sine and cosine functions are the only trigonometric functions for which the domain includes all real numbers.

- Define the domain of sine function

The sine function, \(\text{sin}(x)\), is defined for all real numbers, meaning its domain is \(\text{all real numbers}\).

- Define the domain of cosine function

The cosine function, \(\text{cos}(x)\), is also defined for all real numbers, meaning its domain is \(\text{all real numbers}\).

- Evaluate other trigonometric functions

Functions such as tangent \(\text{tan}(x)\), cotangent \(\text{cot}(x)\), secant \(\text{sec}(x)\), and cosecant \(\text{csc}(x)\) have restrictions. For example, \(\text{tan}(x)\) is not defined when \(\text{cos}(x) = 0\), and \(\text{sec}(x)\) is not defined when \(\text{cos}(x) = 0\). Hence, their domains are not all real numbers.

- Conclusion

Since sine and cosine functions are defined for all real numbers and other trigonometric functions have restrictions, the statement is true.

Key concepts

Domain of Trigonometric Functions

When talking about the domain of trigonometric functions, we are looking at the set of all possible input values (angles) for which the function is defined.
The domain of a function tells us which values of the variable (usually x) we can plug into the function without causing any mathematical issues.
Let's start with the sine and cosine functions.

Sine Function

The sine function is written as \(\text{sin}(x)\).
This function represents the y-coordinate of a point on the unit circle as it moves around the circle.
The sine function is defined for all real numbers. This means that for any angle x you can think of, you will always get a value when you compute sin(x).
This is why we say the domain of the sine function is \(\text{all real numbers}\) or \(\text{−∞, ∞}\).
So, whether x is a whole number, a fraction, or any decimal, the sine function will always give you a valid result.

Cosine Function

The cosine function, written as \(\text{cos}(x)\), represents the x-coordinate of a point on the unit circle.
Like the sine function, the cosine function is defined for all real numbers, meaning you can input any angle x, and it will always return a corresponding value.
Therefore, the domain of the cosine function is also \(\text{all real numbers}\) or \(\text{−∞, ∞}\).
Both sine and cosine functions are unique among the basic trigonometric functions in that they are defined for any input.