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Chapter 10: Problem 92

If \(\sec (\theta)=\frac{x}{4}\) for \(0<\theta<\frac{\pi}{2},\) find an expression for \(\ln |\sec (\theta)+\tan (\theta)|\) in terms of \(x\).

Short Answer

Expert verified
\( \ln \left( \frac{x}{4} \right) + \ln \left( 1 + \sqrt{1 - \frac{16}{x^2}} \right) \)

Step by step solution

01

Recall Trigonometric Identities

Remember that \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Given \( \sec(\theta) = \frac{x}{4} \), we can write \( \cos(\theta) = \frac{4}{x} \).
02

Use Pythagorean Identity to Find Sin(胃)

Use the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Substitute \( \cos(\theta) = \frac{4}{x} \) to get \( \sin^2(\theta) = 1 - \left(\frac{4}{x}\right)^2 = 1 - \frac{16}{x^2} \). Thus, \( \sin(\theta) = \sqrt{1 - \frac{16}{x^2}} \).
03

Express Tan(胃) in Terms of x

Now express \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sqrt{1 - \frac{16}{x^2}}}{\frac{4}{x}} = \frac{x\sqrt{1 - \frac{16}{x^2}}}{4} \).
04

Calculate Sec(胃) + Tan(胃)

The expression \( \sec(\theta) + \tan(\theta) \) will be \( \frac{x}{4} + \frac{x\sqrt{1 - \frac{16}{x^2}}}{4} = \frac{x}{4}(1 + \sqrt{1 - \frac{16}{x^2}}) \).
05

Take the Natural Logarithm

Find \( \ln |\sec(\theta) + \tan(\theta)| \). This simplifies to \( \ln \left( \frac{x}{4}(1 + \sqrt{1 - \frac{16}{x^2}}) \right) = \ln \left( \frac{x}{4} \right) + \ln \left( 1 + \sqrt{1 - \frac{16}{x^2}} \right) \).

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Key Concepts

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

Pythagorean Identity
The Pythagorean identity is a cornerstone of trigonometry, stating that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity helps relate sine and cosine to one another, making it useful for many trigonometric problems.
  • In the given exercise, we use this identity to solve for \( \sin(\theta) \).
  • Given that \( \cos(\theta) = \frac{4}{x} \), substituting it into the identity gives us the equation \( \sin^2(\theta) = 1 - \left( \frac{4}{x} \right)^2 \).
  • This step is crucial in finding \( \tan(\theta) \), a value we need to reach the solution.
This identity simplifies many trigonometric expressions and is widely used in various fields such as physics for wave and oscillation problems.
Secant and Tangent
Secant \( \sec(\theta) \) and tangent \( \tan(\theta) \) are fundamental trigonometric functions originating from sine and cosine.
  • The secant function is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
  • Similarly, the tangent function is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
  • In the exercise, \( \sec(\theta) = \frac{x}{4} \), meaning \( \cos(\theta) = \frac{4}{x} \).
  • Further, \( \tan(\theta) \) can then be expressed in terms of \( x \) using these definitions.
These functions are especially important for describing angles and lengths in triangles and help in transitioning between different trigonometric forms. Understanding their properties is key to mastering this branch of mathematics.
Natural Logarithm
The natural logarithm, often denoted as \( \ln \), is the logarithm to the base \( e \) (where \( e \approx 2.718 \)).
  • Natural logarithms are frequently used to simplify multiplication problems into addition, a core property of logarithms.
  • In the context of this exercise, after finding \( \sec(\theta) + \tan(\theta) \), we apply the logarithm to transform the expression into a more manageable form.
  • The key transformation used is \( \ln(ab) = \ln(a) + \ln(b) \), which simplifies the solution significantly.
Understanding how to manipulate expressions with natural logarithms is invaluable for calculus and higher-level mathematics, where they appear often.
Trigonometric Functions
Trigonometric functions, including sine, cosine, tangent, and secant, describe the relationships between angles and sides in right-angled triangles.
  • The basis of these functions lies in the ratios of the sides of the triangle to one another.
  • For the problem at hand, these functions are used to convert angles into expressions involving \( x \).
  • The exercise demonstrates converting \( \sec(\theta) \) and \( \tan(\theta) \) into calculable expressions using known trigonometric identities and relations.
Grasping these functions allows for a better understanding of geometric principles and aids in solving real-world problems involving cycles and oscillations.

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Most popular questions from this chapter

find the exact value or state that it is undefined. \(\sec \left(\arcsin \left(-\frac{12}{13}\right)\right)\)

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \sin (2 \arcsin (7 x)) $$

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

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Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{11 \pi}{6}\right)\right)\)
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