Problem 109 In Exercises $107-118$, assume... [FREE SOLUTION]

Chapter 10: Problem 109

In Exercises $107-118$, assume that the range of arcsecant is $\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)$ and that the range of arccosecant is $\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]$ when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right) $$

Short Answer

Expert verified

The exact value is $\frac{7\pi}{6}$.

Step by step solution

Determine sec($\frac{5\pi}{6}$)

Evaluate the secant of $\frac{5\pi}{6}$. The secant function is the reciprocal of the cosine function. \[ \sec\left(\frac{5\pi}{6}\right) = \frac{1}{\cos\left(\frac{5\pi}{6}\right)} \]Since $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$, we have:\[ \sec\left(\frac{5\pi}{6}\right) = -\frac{2}{\sqrt{3}} \]

Simplify the Secant Value

Simplify the value of $-\frac{2}{\sqrt{3}}$ by rationalizing the denominator:\[ \sec\left(\frac{5\pi}{6}\right) = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \]

Determine the Arcsecant Range

We need the range for $\operatorname{arcsec}$, which is $\left[0, \frac{\pi}{2}\right) \cup \left[\pi, \frac{3\pi}{2}\right)$. Since our result needs to fall within this range, identify any standard angle value in these ranges whose secant matches the calculated value.

Find $\operatorname{arcsec}$ Value

Identify the angle whose secant is $-\frac{2\sqrt{3}}{3}$ and falls within the specified range for $\operatorname{arcsec}$. Since $\sec\left(\frac{5\pi}{6}\right) = -\frac{2\sqrt{3}}{3} $ and $\frac{5\pi}{6}$ initially falls outside the first section of the specified $\arcsec$ range, we must convert it using a coterminal angle in the $\arcsec$ range. Our angle calculation coincides with the angle $\frac{5\pi}{6}$ which falls within $\left[\pi, \frac{3\pi}{2}\right)$ when calculated as $\frac{7\pi}{6}$. Therefore, the correct value that satisfies $\operatorname{arcsec}(\sec(\theta)) = \theta$ is actually $\frac{7\pi}{6}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arcsecant Function

The arcsecant function, often denoted as $ \operatorname{arcsec}(x) $, is the inverse of the secant function. This means it helps you find the angle whose secant is a given number. Here's a breakdown:

The secant of an angle, say $ \theta $, is the reciprocal of its cosine. So, $ \sec(\theta) = \frac{1}{\cos(\theta)} $.
The arcsecant function reverses this. If $ \theta = \operatorname{arcsec}(x) $, then $ x = \sec(\theta) $.
The arcsecant has a specified range: $ \left[0, \frac{\pi}{2}\right) \cup \left[\pi, \frac{3\pi}{2}\right) $. Only angles within this range are considered.

This range avoids ambiguous results and focuses on principal angles. When evaluating $ \operatorname{arcsec} $, ensure the result angle fits within this set range.

Secant Function

The secant function, denoted as $ \sec $, is one of the more complex trigonometric functions but is foundational in trigonometry. Here's how it works:

It is defined as the reciprocal of the cosine function. Thus, $ \sec(\theta) = \frac{1}{\cos(\theta)} $.
Unlike sine or cosine, which never exceed $-1$ or $1$, the secant can be any real number except values between $-1$ and $1$.
For angles commonly measured in radians, such as $ \frac{5\pi}{6} $, secant values are significant since they relate directly to cosine values.

In practical scenarios, the secant function allows us to compute angles' properties that lie on the unit circle. It's particularly useful when dealing with the geometric aspects of trigonometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable within a certain range. These identities are crucial for simplifying expressions and solving equations involving angles.

The reciprocal identities: These identities include $ \sec(\theta) = \frac{1}{\cos(\theta)} $, which is fundamental to understanding how the secant function operates.
Pythagorean identities: Essential for relating sine, cosine, and tangent functions, such as $ \sin^2(\theta) + \cos^2(\theta) = 1 $.
Angle addition identities: Which include formulas like $ \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) $.

These identities help in transforming trigonometric expressions into simpler forms, often revealing relationships like coterminal angles. Recognizing these patterns is key to solving tricky trigonometry problems.

Angle Measurement in Radians

Angles in trigonometry can be measured in radians or degrees, but radians are particularly favored in higher mathematics because they naturally relate to the unit circle.

A full circle in radians is $ 2\pi $, which correlates to $ 360^\circ $ in degrees. Thus, $ \pi $ radians equal $ 180^\circ $.
Many key angles, such as $ \frac{5\pi}{6} $, are often expressed in radians because they simplify calculus operations like differentiation and integration.
Knowing how to convert between radians and degrees is essential: Multiply degrees by $ \frac{\pi}{180} $ to convert to radians.

Radians provide a natural framework for solving problems involving trigonometric functions and their inverses, offering more straightforward manipulation in equations and identities.

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Short Answer

Step by step solution

Determine sec(\(\frac{5\pi}{6}\))

Simplify the Secant Value

Determine the Arcsecant Range

Find \(\operatorname{arcsec}\) Value

Key Concepts

Arcsecant Function

Secant Function

Trigonometric Identities

Angle Measurement in Radians

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