Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \ln (\sin x)=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{2} + 2k\pi \).

Step 1: Understanding the problem

The given equation is \( \ln(\sin x) = 0 \). The question asks us to find all values of \( x \) that satisfy this equation.

Step 2: Removing the logarithm

To find the solutions, we need to remove the logarithm. Since \( \ln(a) = 0 \) implies \( a = 1 \), we set \( \sin x = 1 \).

Step 3: Finding the general solution

The sine function reaches the value of 1 at specific angles. Recall that \( \sin x = 1 \) at \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is any integer, because sine is a periodic function with period \( 2\pi \).

Step 4: Final expression of solutions

Thus, the solutions of the equation are \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer. This includes angles such as \( \frac{\pi}{2}, \frac{5\pi}{2}, \frac{9\pi}{2}, \ldots \).

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry. It's typically defined for an angle in a right triangle, but more commonly extended to any real number, corresponding to an angle measuring that number in radians. The sine of an angle is the vertical coordinate of the corresponding point on the unit circle.

Some important characteristics of the sine function include:

• Range: The sine function can only take values between -1 and 1, inclusive.
• Zeros: It is zero at integer multiples of π (i.e., 0, π, 2π, 3π, etc.).
• Maximum Value: It reaches its maximum value of 1 at \(x = \frac{\pi}{2} + 2k\pi\).
• Minimum Value: It reaches its minimum of -1 at \(x = \frac{3\pi}{2} + 2k\pi\).

Understanding these properties is crucial for solving equations that involve the sine function, like finding values of \(x\) such that \(\sin(x) = 1\) as seen in our original exercise.

General Solution

When solving trigonometric equations, we often seek not just a single solution, but all possible solutions. This is due to the periodic nature of trigonometric functions. The general solution accounts for this periodicity and gives us a formula from which we can derive all specific solutions.

In our original problem, after simplifying \(\ln(\sin x) = 0\) to \(\sin x = 1\), we need the general solution that covers all angles where sine reaches 1.

The general solution for \(\sin x = 1\) is \(x = \frac{\pi}{2} + 2k\pi\), where \(k\) is an integer. This accounts for the repetitive nature of sine, repeating every \(2\pi\) radians.

This form helps us see the infinite sequence of solutions such as \(\frac{\pi}{2}\), \(\frac{5\pi}{2}\), \(\frac{9\pi}{2}\), and so forth, capturing the elemental cycle of the sine function.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals. The sine function is a classic example of a periodic function.

Key characteristics of periodic functions include:

• Period: The period is the smallest interval over which the function repeats. For the sine function, this period is \(2\pi\).
• Symmetry: Sine exhibits odd symmetry, meaning \(\sin(-x) = -\sin(x)\).
• Repetition: Given its period, if \(\sin(x) = a\), then \(\sin(x + 2k\pi) = a\) for any integer \(k\).

These properties are essential because they enable the simplification of complex trigonometric equations by understanding that the function behaves predictably over regular intervals. This predictability was used in our original equation to determine that the sine function's solutions repeat every \(2\pi\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved.

Some key identities useful in solving trigonometric equations include:

• Basic Identities: These include functions like \(\sin^2(x) + \cos^2(x) = 1\), illustrating relationships between sine and cosine.
• Cofunction Identities: Such as \(\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\), providing connections between the different functions.
• Periodic Identities: For example, \(\sin(x + 2\pi) = \sin(x)\), demonstrating the foundational periodic nature of sine and cosine.

These identities are incredibly powerful tools when manipulating and solving equations involving trigonometric functions. They provide relationships that can simplify the process, helping to isolate variables and find solutions efficiently as in our original example.