Problem 109 - Precalculus

Question

Most calculators do not have a key to evaluate sec \(^{-1}(2) .\) Explain how this can be done using the cosine function or the inverse cosine function.

Step-by-Step Solution

Answer

\( \sec^{-1}(2) = \frac{\pi}{3} \) radians, which is 60°.

Step 1: Understanding the Relationship between Secant and Cosine

The secant function is the reciprocal of the cosine function. Therefore, given \( \sec^{-1}(2) \), we know that \( \sec(\theta) = 2 \) implies \( \cos(\theta) = \frac{1}{2} \).

Step 2: Using Inverse Cosine

We use the inverse cosine function to find \( \theta \). Since \( \cos(\theta) = \frac{1}{2} \), we have \( \theta = \cos^{-1}\left(\frac{1}{2}\right) \).

Step 3: Evaluating the Inverse Cosine Function

The value of \( \cos^{-1}\left(\frac{1}{2}\right) \) is a known value in trigonometry. \( \theta = \cos^{-1}\left(\frac{1}{2}\right) \) is equal to \( \frac{\pi}{3} \) radians (or 60°).

Step 4: Interpreting the Result

Hence, \( \sec^{-1}(2) \) is \( \frac{\pi}{3} \) radians or 60°.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(\theta) \), is one of the six primary trigonometric functions. It is particularly significant because it is the reciprocal of the cosine function. In simpler terms, while the cosine of an angle gives the ratio of the adjacent side over the hypotenuse in a right triangle, the secant function provides the inverse of this ratio.

This relationship can be expressed as:

• The cosine function: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \)
• The secant function: \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent side}} \)

Understanding that secant is the reciprocal of cosine is key when working with problems that involve secant, especially when your calculator lacks a direct secant function key.

By redefining problems in terms of cosine, you can effectively work through exercises involving secant without directly computing it.

Cosine Function

The cosine function is one of the foundational trigonometric functions. Not only is it pivotal for understanding angles in a triangle, but it also plays a crucial role in converting between different trigonometric expressions. In the context of inverse problems, the cosine function allows us to find angles using known sides of a triangle.

Consider an instance where you need to solve \( \sec^{-1}(2) \). Instead of working directly with secant, you utilize the fact that \( \cos(\theta) = \frac{1}{2} \) when \( \sec(\theta) = 2 \). Thus, knowing the cosine function can transform potentially complex calculations into straightforward ones by using inverse cosine, or \( \cos^{-1} \), to find the angle \( \theta \).

For example, when \( \cos^{-1}(\frac{1}{2}) \) is evaluated, it yields an angle of \( \frac{\pi}{3} \) radians, highlighting its usefulness in addressing inverse trigonometric problems.

Reciprocal Relationships

Reciprocal relationships between trigonometric functions are foundational concepts in understanding how these functions relate to one another. By definition, a reciprocal relationship means that two functions are inverses in terms of multiplication: the product of the functions equals one.

For trigonometry:

• Secant and cosine are reciprocals: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Other examples include sine and cosecant, as well as tangent and cotangent.

Because of these relationships, you can convert between these functions easily. For instance, knowing \( \sec^{-1}(2) \) requires converting to cosine by noting that \( \sec(\theta) = 2 \) implies \( \cos(\theta) = \frac{1}{2} \). You then solve using the inverse cosine function.

This reciprocal relationship transforms the problem into a format that can be computed, especially useful when working with available calculator functions or solving theoretical problems.