Determine whether each statement is possible or impossible. a. sec θ = ―2/3

Verified step by step guidance

1. Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\sec \theta\) is the reciprocal of \(\cos \theta\).
2. Since \(\cos \theta\) ranges between \(-1\) and \$1$ for all real values of \(\theta\), the values of \(\sec \theta\) must satisfy \(|\sec \theta| \geq 1\) or be undefined where \(\cos \theta = 0\).
3. Check the given value \(\sec \theta = -\frac{2}{3}\). The absolute value is \(\left| -\frac{2}{3} \right| = \frac{2}{3}\), which is less than 1.
4. Because \(|\sec \theta|\) must be greater than or equal to 1, \(\sec \theta = -\frac{2}{3}\) is not possible for any real angle \(\theta\).
5. Therefore, the statement \(\sec \theta = -\frac{2}{3}\) is impossible.

Key Concepts

Definition and Range of Secant Function

The secant function, sec θ, is defined as the reciprocal of the cosine function: sec θ = 1/cos θ. Since cosine values range between -1 and 1, secant values are either greater than or equal to 1 or less than or equal to -1. Values between -1 and 1 are not possible for sec θ.

Reciprocal Relationship Between Secant and Cosine

Because sec θ = 1/cos θ, if sec θ is given, the corresponding cosine value can be found by taking its reciprocal. This relationship helps determine if a given secant value is possible by checking if the reciprocal lies within the cosine function's valid range.

Determining Possibility of Trigonometric Values

To determine if a trigonometric value is possible, compare it against the function's range. For sec θ, values must satisfy |sec θ| ≥ 1. If a given value falls outside this range, it is impossible for any angle θ to produce that value.