Problem 73 - Precalculus with Calculus Previews

Question

In Problems \(73-76\), verify the given identity. $$ \ln |\sec x|=-\ln |\cos x| $$

Step-by-Step Solution

Answer

The identity is verified by using the property of logarithms and the reciprocal identity of secant.

Step 1: Recall Logarithm Properties

Recall that for any positive number, the natural logarithm has the property: \( \ln |a| = - \ln |b| \) if \( a = \frac{1}{b} \).

Step 2: Express \( \sec x \) in terms of \( \cos x \)

The secant function is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). Hence, \(|\sec x| = \left| \frac{1}{\cos x} \right| = \frac{1}{|\cos x|}\).

Step 3: Substitute and Apply Logarithm Properties

Substitute the expression for \(|\sec x|\) into the logarithm identity:\[\ln \left| \frac{1}{\cos x} \right| = -\ln |\cos x|\]Using the property \( \ln \left( \frac{1}{a} \right) = -\ln(a) \), it follows that: \( \ln \left( \frac{1}{|\cos x|} \right) = - \ln |\cos x| \).

Step 4: Conclusion

The identity \( \ln |\sec x| = - \ln |\cos x| \) holds true based on the properties of logarithms and the reciprocal identity of trigonometric functions.

Key Concepts

Logarithm Properties

Logarithms are powerful mathematical tools that help simplify multiplication and division into addition and subtraction. This transformation is particularly useful when dealing with large numbers. A key property of logarithms is the logarithm of a reciprocal. For any positive number \( a \), if \( a \) equals \( \frac{1}{b} \), then the property \( \ln |a| = - \ln |b| \) holds. Essentially, the natural logarithm of a reciprocal is the negative of the natural logarithm of the number itself. This rule simplifies our calculations when evaluating expressions like \( \ln \left( \frac{1}{a} \right) \) to \(- \ln(a) \), which was applied in the solution of the exercise provided.

Reciprocal Trigonometric Functions

The trigonometric functions have reciprocal counterparts that relate them in a special way. The main reciprocal trigonometric functions are:

• Cosecant (\( \csc x \)) is the reciprocal of sine (\( \sin x \)), where \( \csc x = \frac{1}{\sin x} \).
• Secant (\( \sec x \)) is the reciprocal of cosine (\( \cos x \)), where \( \sec x = \frac{1}{\cos x} \).
• Cotangent (\( \cot x \)) is the reciprocal of tangent (\( \tan x \)), where \( \cot x = \frac{1}{\tan x} \).

Understanding these relationships helps in manipulating and simplifying trigonometric expressions, which is critical in verifying identities and solving trigonometric equations. In the exercise, \( \sec x \) is expressed in terms of \( \cos x \) using its reciprocal nature, making it straightforward to apply logarithmic properties for simplification.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the study of periodic phenomena. The primary trigonometric functions include:

• Sine (\( \sin x \))
• Cosine (\( \cos x \))
• Tangent (\( \tan x = \frac{\sin x}{\cos x} \))

These functions arise from the ratios of sides of a right-angled triangle or, more generally, from the coordinates of points on a unit circle in trigonometry. They are periodic, with specific patterns repeating over regular intervals. For instance, both sine and cosine functions have a period of \(2\pi\). Their reciprocal functions, such as secant and cosecant, also share this periodic nature and are used in various identities and equations.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm with a base \(e\), where \(e\) is approximately equal to 2.71828. Natural logarithms are important in many areas of mathematics and science because they simplify the process of finding derivatives and integrals in calculus. The natural logarithm has key properties similar to other logarithms:

• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• \(\ln(a^b) = b \ln(a)\)

These properties allow us to break down complex logarithmic expressions into more manageable parts, helping in verifying identities like \( \ln |\sec x| = - \ln |\cos x| \). Understanding these concepts is vital for tackling problems involving logarithmic and trigonometric identities effectively.