Problem 38 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$\ln (\sin x)=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.

Step 1: Understand the Equation

The given equation is \( \ln(\sin x) = 0 \). According to the properties of logarithms, if \( \ln(a) = 0 \), then \( a = 1 \). Thus, we need \( \sin x = 1 \).

Step 2: Solve \( \sin x = 1 \)

For the sine function, \( \sin x = 1 \) occurs when \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is any integer. This is because the sine function has a period of \( 2\pi \), and it reaches the value of 1 at every half of its period starting from \( \frac{\pi}{2} \).

Step 3: Write the General Solution

The general solution for the equation is \( x = \frac{\pi}{2} + 2k\pi \). This formula provides all solutions to the original equation by substituting integer values for \( k \).

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics, often used to solve equations involving exponents. In simple terms, the logarithm of a number is the exponent by which another fixed number, the base, has to be raised to produce that number. For example, if we have the equation \( a = b^c \), then \( \log_b(a) = c \). Logarithms are extremely useful in solving equations where the variable appears as an exponent.

When dealing with natural logarithms, denoted as \( \ln \), the base is \( e \), where \( e \approx 2.718 \). The natural logarithm \( \ln(x) \) is the power to which \( e \) must be raised to get \( x \). An important property to remember is that \( \ln(1) = 0 \). This property is crucial when solving equations like \( \ln(\sin x) = 0 \), as you can interpret this to mean that \( \sin x \) must equal 1 since \( \ln(1) = 0 \). Such properties simplify the solving process in logarithmic equations.

Sine Function

The sine function is one of the primary trigonometric functions, commonly abbreviated as \( \sin \). It is a periodic function, meaning that it repeats its values at regular intervals. The sine function in terms of angle \( x \) gives the measure of the y-coordinate of a point on the unit circle.

• Key Values: The sine function achieves its peak value of 1 when \( x = \frac{\pi}{2} \), and it repeats this peak every \( 2\pi \) radians.
• Periodicity: The function is periodic with a period of \( 2\pi \), meaning \( \sin(x) = \sin(x + 2k\pi ) \) for any integer \( k \).
• Positivity: The sine function is positive in the first and second quadrants of the unit circle.

In the context of the equation \( \sin x = 1 \), this tells us that solutions occur at \( x = \frac{\pi}{2} + 2k\pi \), because the sine function equals 1 at the angle \( \frac{\pi}{2} \) and every full rotation afterwards due to its periodic nature.

General Solutions

In trigonometry, a general solution provides all possible solutions to an equation, considering the periodic nature of the trigonometric functions. When we refer to the general solution of the equation \( \ln(\sin x) = 0 \), we derive it through understanding the sine function’s behavior and characteristics.

The general solution is expressed as \( x = \frac{\pi}{2} + 2k\pi \), with \( k \) being an integer. This formula accounts for:

• The periodicity of the sine function, ensuring we include all instances where the value of sine returns to 1.
• The property of logarithms where \( \ln(1) = 0 \), constraining us to find sine values that equal 1.

To solve and write these kinds of equations efficiently:

• Recognize the trigonometric function's key behavior and its cycle.
• Use the basic properties related to the logarithmic function.

This allows for comprehensive solutions that encapsulate all possible values of \( x \) in the given context, making it practical for understanding and solving problems.