Problem 27 - Thomas Calculus

Question

Find the values in Exercises \(17-28\) $$ \sec ^{-1}\left(\sec \left(-\frac{\pi}{6}\right)\right) \quad \text { (The answer is } \text {not}-\pi / 6 . ) $$

Step-by-Step Solution

Answer

The value is \( \frac{\pi}{6} \).

Step 1: Understand the Inverse Function

The problem involves the inverse secant function \( \sec^{-1} \), which returns an angle whose secant is the given value. Here, we want to find \( \sec^{-1}\left(\sec\left(-\frac{\pi}{6}\right)\right) \). The secant function, \( \sec(\theta) = \frac{1}{\cos(\theta)} \), is defined except where \( \cos(\theta) = 0 \).

Step 2: Calculate \( \sec(-\frac{\pi}{6}) \)

First, find \( \sec\left(-\frac{\pi}{6}\right) \). Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we calculate \( \cos\left(-\frac{\pi}{6}\right) \). Using the cosine function, \( \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \). Thus, \( \sec\left(-\frac{\pi}{6}\right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).

Step 3: Apply the Inverse Function

Next, apply the inverse secant function: \( \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \). The range of \( \sec^{-1}(x) \) is \( [0, \pi] \) excluding \( \frac{\pi}{2} \), but we need a value that corresponds to the calculated secant value in the principal range.

Step 4: Verify the Principal Angle

The principal angle for \( \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \) that gives the correct secant value within the range \( [0, \pi] \) is \( \frac{\pi}{6} \). Since the cosine function is even, \( \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) \), thus \( \sec(-\frac{\pi}{6}) \) has the same magnitude as \( \sec(\frac{\pi}{6}) \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec(\theta) \), is the reciprocal of the cosine function. It is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This function helps in understanding the ratio of the hypotenuse to the adjacent side in a right triangle.
The key points about secant are:

• It is undefined wherever the cosine of an angle is zero.
• It relates directly to geometry, specifically in right-angled triangles.

This function tends to have larger values as the cosine function decreases, which can lead to comprehending how angles behave when extended beyond the right angles.

Inverse Function

Inverse trigonometric functions bring us back from a ratio to an angle. When discussing the inverse secant function, \( \sec^{-1}(x) \), we seek an angle whose secant is \( x \).
Some characteristics of inverse functions include:

• They reverse the effect of the original trigonometric function.
• For \( \sec^{-1}(x) \) specifically, the angle returned falls within a specific range.
• They are crucial for solving equations where the angle is the unknown element.

Understanding these concepts aids in determining angles from given trigonometric values accurately and appropriately.

Principal Values

Principal values are particularly important in inverse trigonometric functions, ensuring the results fall within a standard range. For the inverse secant function, the principal value is restricted to the angle within \([0, \pi]\), excluding \(\frac{\pi}{2}\).
Principal values ensure:

• Unambiguous and consistent results across calculations.
• That these values are within a generally accepted range for mathematical operations.

By using principal values, students identify angles in a uniform manner, alleviating confusion while dealing with inverse trigonometric functions.

Cosine Function

The cosine function is foundational in understanding the secant function, as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This function provides information about the x-coordinate on the unit circle, representing the ratio of the adjacent side to the hypotenuse.
Key aspects of the cosine function are:

• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• Its range is bounded between -1 and 1.
• It helps in graphically representing wave patterns.

Understanding cosine is vital in trigonometry, facilitating comprehension of periodic phenomena and their mathematical expressions.