Question

For what numbers \(\theta\) is \(f(\theta)=\sec \theta\) not defined?

Short answer

The function \( \sec \theta \) is not defined for \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is any integer.

Step-by-step solution

Recall the Definition of Secant Function

The secant function is defined as the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \)This means the function \( \sec \theta \) will not be defined when \( \cos \theta = 0 \).

Determine When Cosine is Zero

Next, identify the angles where the cosine of \( \theta \) is zero. The cosine function, \( \cos \theta \), is zero at \( \theta = \frac{\pi}{2} + k\pi \) for any integer \( k \).

Summarize the Angles

Combine the information and conclude that \( \sec \theta \) is not defined for \( \theta = \frac{\pi}{2} + k\pi \) where \( k \) is any integer.

Key concepts

trigonometric functions

Trigonometric functions are fundamental in mathematics and are linked with the angles of triangles and periodic phenomena. There are six main trigonometric functions, which include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions help us relate the angles of a right triangle to the lengths of its sides.
In this exercise, we are particularly interested in the secant function, which is the reciprocal of the cosine function. Understanding how these functions interrelate is key to solving trigonometric problems.

cosine function

The cosine function is one of the primary trigonometric functions and is denoted as \( \cos \theta \). It represents the ratio of the adjacent side to the hypotenuse in a right triangle.
The cosine function is crucial in trigonometry and plays a significant role in defining other trigonometric functions such as secant (sec) which is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
The cosine function has specific points where it equals zero. These points are particularly important for determining when the secant function is undefined. Specifically, the cosine of an angle is zero at \( \theta = \frac{\pi}{2} + k\pi \) where \( k \) is any integer. At these angles, the value of the secant function becomes undefined.

undefined values in trigonometry

In trigonometry, certain functions can become undefined when their values result in a division by zero. For the secant function, given by \( \sec \theta = \frac{1}{\cos \theta} \), it's evident that secant is undefined whenever \( \cos \theta \) is zero.
The cosine function is zero at specific angles, which are given by \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is any integer. At these angles, the division by zero in the secant function causes it to be undefined.
To summarize, the secant function \( \sec \theta \) is not defined wherever \( \cos \theta = 0 \), corresponding to \( \theta = \frac{\pi}{2} + k\pi \). Understanding these points helps to identify and avoid undefined values in trigonometric functions.