Decide whether each statement is true or false. If false, explain why.
The tangent and secant functions are undefined for the same values.

Verified step by step guidance

1. Identify the definitions of the tangent and secant functions in terms of sine and cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
2. Determine when the tangent function is undefined: \( \tan(\theta) \) is undefined when \( \cos(\theta) = 0 \) because division by zero is undefined.
3. Determine when the secant function is undefined: \( \sec(\theta) \) is also undefined when \( \cos(\theta) = 0 \) for the same reason.
4. Identify the values of \( \theta \) where \( \cos(\theta) = 0 \). These occur at odd multiples of \( \frac{\pi}{2} \) (e.g., \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)).
5. Conclude that both tangent and secant functions are undefined for the same values of \( \theta \), specifically where \( \cos(\theta) = 0 \). Therefore, the statement is true.

Key Concepts

Tangent Function

The tangent function, defined as the ratio of the sine to the cosine of an angle (tan(θ) = sin(θ)/cos(θ)), is undefined when the cosine of the angle is zero. This occurs at odd multiples of π/2 (90 degrees), where the function approaches infinity, leading to vertical asymptotes on the graph.

Secant Function

The secant function is the reciprocal of the cosine function (sec(θ) = 1/cos(θ)). It is undefined at the same angles where the cosine is zero, specifically at odd multiples of π/2 (90 degrees). Thus, secant also has vertical asymptotes at these points, indicating that the function does not have a defined value.

Undefined Functions

A function is considered undefined at certain points when it cannot produce a valid output. For both tangent and secant functions, this occurs at angles where the denominator of their respective ratios (cosine for tangent and secant) equals zero. Understanding these undefined points is crucial for analyzing the behavior of these trigonometric functions.