Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms and trigonometric identities to verify the identity. $$\ln |\sec \theta|=-\ln |\cos \theta|$$

Step-by-Step Solution

Answer

The given identity \( \ln |\sec \theta|=-\ln |\cos \theta| \) can be verified by rewriting secant in terms of cosine and applying the logarithm's rule that \( \ln \frac{1}{a} \) is equivalent to \( -\ln {a} \).

Step 1: Write sec in terms of cosine

In terms of cosine, secant \( (\sec \theta) \) is defined as \( \frac{1}{\cos \theta} \). So, we can rewrite \( \ln |\sec \theta| \) as \( \ln |\frac{1}{\cos \theta}| \).

Step 2: Apply the property of Logarithms

Next, we apply the property of logarithms. According to the logarithm rules, \( \ln \frac{1}{a} \) is equivalent to \( -\ln {a} \). Hence, \( \ln |\frac{1}{\cos \theta}| \) can be rewritten as \( -\ln |\cos \theta| \).

Step 3: Confirm the equality

It can now be seen that \( \ln |\sec \theta|=-\ln |\cos \theta| \), hence verifying the given identity.

Key Concepts

Properties of Logarithms

Logarithms help us simplify complex multiplications and divisions by converting them into addition and subtraction. This property makes logarithms extremely powerful in mathematical calculations. One fundamental property is the use of the inverse. In the context of logarithms, the inverse property tells us that the logarithm of a reciprocal can be expressed as the negative of the logarithm. Specifically, for a non-zero number \( a \), \( \ln \left( \frac{1}{a} \right) = -\ln (a) \).

This property is used extensively when simplifying logarithmic expressions. For instance, if you have \( \ln \left( \frac{1}{\cos \theta} \right) \), you can apply this property to get \( -\ln (\cos \theta) \). This conversion is advantageous as it simplifies the handling of inverse trigonometric functions in logarithmic applications.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all the values of the variables in their domains. These identities are vital as they provide tools for simplifying and solving trigonometric equations. There are several fundamental categories of trigonometric identities:

• Reciprocal Identities: These identities show the relationships between sine, cosine, tangent, and their respective reciprocals, cosecant, secant, and cotangent.
• Pythagorean Identities: These involve squares of the sine, cosine, and tangent functions to show their inherent relationship in a right-angle triangle.

In our exercise, we specifically use the reciprocal identity, which defines \( \sec \theta \) as \( \frac{1}{\cos \theta} \). This identity is the key to transforming \( \ln |\sec \theta| \) into a more manageable form that can be expressed using cosine.

Secant Function

The secant function is one of the six fundamental trigonometric functions that arise from the geometry of triangles. Its symbol is \( \sec \theta \), and it is defined as the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \).

The secant function is particularly important in trigonometry and calculus because it simplifies computations involving division by the cosine. It plays a crucial role in scenarios where direct calculation with cosine would be cumbersome. In the logarithmic context, taking the logarithm of the secant translates into a division involving cosine, as shown by the identity \( \ln |\sec \theta| = -\ln |\cos \theta| \). Through this transformation, we can leverage our understanding of the properties of logarithms to make sense of more complex trigonometric expressions.

The secant function not only aids in simplifying trigonometric identities but also enables the verification and proof of mathematical statements by merging trigonometric transformations with logarithmic properties.