Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

Yes, \( \ln |\cot \theta| = \ln |\cos \theta| - \ln |\sin \theta| \) is verified by applying logarithmic properties and trigonometric identities.

Step 1: Apply Logarithm Property

Using the logarithmic property \( \ln(a \div b) = \ln a - \ln b \), rewrite \( \ln |\cos \theta| - \ln |\sin \theta| \) as \( \ln |\frac{\cos \theta}{\sin \theta}| \).

Step 2: Use Trigonometric Identity

Simplify \( \ln |\frac{\cos \theta}{\sin \theta}| \) using the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, we have \( \ln |\cot \theta| \).

Step 3: Confirm the Identity

Both sides of the equation now match. This confirms that \( \ln |\cot \theta| = \ln |\cos \theta| - \ln |\sin \theta| \).

Key Concepts

Properties of Logarithms

Logarithms are mathematical operations that are, in a sense, the inverse of exponentiation. They have unique properties that allow for the simplification and solving of complex equations. One key property is the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator, expressed as \( \text{log}_b \frac{a}{c} = \text{log}_b a - \text{log}_b c \).

Other vital properties include the product rule and the power rule. The product rule says that the logarithm of a product is equal to the sum of the logarithms of the individual factors (\( \text{log}_b (ac) = \text{log}_b a + \text{log}_b c \)), while the power rule indicates that the logarithm of a power is equal to the exponent times the logarithm of the base (\( \text{log}_b a^c = c \times \text{log}_b a \)). These properties are essential tools in both simple and complex calculations involving logarithms.

Trigonometric Identities Verification

Verifying trigonometric identities is a fundamental concept in trigonometry involving proving that two expressions are equivalent for all values of the variables involved. To verify an identity, one often starts with one side of the equation and manipulates it using known trigonometric identities until it matches the other side.

Key identities used in these verifications include, but are not limited to, the Pythagorean identities, the co-function identities, and the angle sum and difference identities. Proving an identity might require a combination of algebraic manipulation and strategic use of these identities to simplify the expressions to their most basic form. The verification process not only confirms the equality but also strengthens understanding of the interrelations between trigonometric functions.

Cotangent Identity

In trigonometry, the cotangent function is one of the six primary trigonometric functions and is the reciprocal of the tangent function. The cotangent identity is defined as \( \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \). This identity is often utilized in solving trigonometric equations and proving other identities.

Understanding the cotangent identity is crucial for simplifying complex trigonometric expressions. Since cotangent is less commonly used than sine or cosine, reexpressing it in terms of these better-understood functions can make the problem more approachable. Remembering that the cotangent of an angle in a right triangle represents the ratio of the adjacent side to the opposite side will also help in visualizing and solving trigonometric problems.

Logarithmic Properties Application

The application of logarithmic properties is not limited to theoretical exercises; it plays a significant role in solving real-world problems across various scientific and engineering disciplines. For example, when the properties of logarithms are applied to decibel calculations in acoustics or to the Richter scale for measuring earthquake intensity, complex multiplicative processes can be transformed into more manageable additive ones.

In the context of verifying trigonometric identities, these properties allow us to rewrite expressions in simpler forms, which can make it easier to see relationships between different trigonometric functions. Accurate application of these properties requires practice and a good understanding of their rationale. Remember that logarithms can only be applied to positive real numbers, which is why absolute values are often used in conjunction with logarithmic functions to ensure the arguments are within the domain of definition.