Problem 53 - Precalculus

Question

Determine whether each statement is possible or not. $$\sec \theta=-\frac{4}{\sqrt{7}}$$

Step-by-Step Solution

Answer

The statement is possible because \( \cos \theta = -\frac{\sqrt{7}}{4} \) falls within \([-1, 1]\).

Step 1: Understand Secant Function

The secant function is defined as the reciprocal of the cosine function:\[ \sec \theta = \frac{1}{\cos \theta} \]To understand if the given statement is possible, we need to analyze the implication on the cosine function.

Step 2: Reciprocal of Secant and Cosine Range

Since \( \sec \theta = -\frac{4}{\sqrt{7}} \), we find that:\[ \cos \theta = \frac{1}{\sec \theta} = -\frac{\sqrt{7}}{4} \]The cosine function ranges from -1 to 1. We must check if \( -\frac{\sqrt{7}}{4} \) lies within this range.

Step 3: Simplify and Compare Values

Calculate \( \sqrt{7} \) approximately which is roughly 2.645. Therefore, \( -\frac{\sqrt{7}}{4} \approx -\frac{2.645}{4} \approx -0.66125 \).Since -0.66125 is within the range \([-1, 1]\), the value \( \cos \theta \) is indeed possible.

Step 4: Conclusion

The analysis shows that the value \( \cos \theta = -\frac{\sqrt{7}}{4} \) falls within the valid range of the cosine function.Therefore, \( \sec \theta = -\frac{4}{\sqrt{7}} \) is possible since it leads to a valid cosine value.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is an important trigonometric function. It is defined as the reciprocal of the cosine function. This means that if you know \( \cos \theta \), you can easily find \( \sec \theta \) using the formula:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Let's understand this reciprocal nature more clearly. If \( \cos \theta \) is the cosine value of an angle \( \theta \), then \( \sec \theta \) gives you the inverse relationship. This relationship makes the secant function particularly useful in various geometric and analytical contexts. The key concept to keep in mind about secant is that it puts focus on angles where cosine is defined but revamps them in an inverse context. The secant function can have very large values, as it increases significantly when cosine values approach zero. This property is used in the original problem to determine the possibility of a negative secant value.

Cosine Function Range

The cosine function, represented as \( \cos \theta \), has a well-defined range that you must know. Let's uncover what this range is and why it's significant. For the cosine function, the values it can take are always between -1 and 1. This restriction on its range means:

• The lowest value \( \cos \theta \) can attain is -1.
• The highest value it can reach is 1.

These boundaries are crucial when dealing with trigonometric equations and inequalities because they tell you what values are achievable within a real context. When the problem asks about the possibility of \( \cos \theta = -\frac{\sqrt{7}}{4} \), referring back to the range quickly assures us if such a value lies within the permissible interval. Using approximations can help verify whether specific values fall within the maximum and minimum limits. In the example, calculating \( -\frac{\sqrt{7}}{4} \) as approximately -0.66125, confirmed that it indeed lies between -1 and 1, making it a valid cosine value.

Reciprocal Trigonometric identities

Trigonometric identities are like a toolkit in mathematics. Among these, reciprocal identities are particularly helpful when solving problems involving the secant function and others. Let’s explore what these are. Reciprocal trigonometric identities relate each of the primary trigonometric functions to each other in terms of their reciprocals. Here are the basic ones you should know:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities allow you to express angles expressed in one trigonometric function in terms of another, through reciprocation. This is immensely useful in simplifying complex trigonometric expressions or solving equations, as demonstrated in the original problem. Here, the given secant value \( \sec \theta = -\frac{4}{\sqrt{7}} \) was translated into the corresponding cosine value \( \cos \theta = -\frac{\sqrt{7}}{4} \) by applying the reciprocal identity. This method simplifies checking the validity of trigonometric expressions by leveraging limits imposed by each function's natural range. Understanding and applying these reciprocal identities can greatly enhance your ability to solve trigonometric equations efficiently.