Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \sin \left(2 x-\frac{\pi}{3}\right)=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{4} + k\pi \) and \( x = \frac{7\pi}{12} + k\pi \), where \( k \) is an integer.

Step 1: Set the Trigonometric Equation

We start solving the equation \( \sin \left(2x - \frac{\pi}{3}\right) = \frac{1}{2} \). We know that \( \sin(\theta) = \frac{1}{2} \) at angles \( \theta = \frac{\pi}{6} + 2k\pi \) and \( \theta = \frac{5\pi}{6} + 2k\pi \), where \( k \) is an integer.

Step 2: Solve for x using the First Solution for θ

Set \( 2x - \frac{\pi}{3} = \frac{\pi}{6} + 2k\pi \) and solve for \( x \). Add \( \frac{\pi}{3} \) to both sides to get: \( 2x = \frac{\pi}{6} + \frac{\pi}{3} + 2k\pi \) Combine fractions: \( 2x = \frac{\pi}{6} + \frac{2\pi}{6} + 2k\pi = \frac{3\pi}{6} + 2k\pi = \frac{\pi}{2} + 2k\pi \)Divide both sides by 2: \( x = \frac{\pi}{4} + k\pi \)

Step 3: Solve for x Using the Second Solution for θ

Set \( 2x - \frac{\pi}{3} = \frac{5\pi}{6} + 2k\pi \) and solve for \( x \).Add \( \frac{\pi}{3} \) to both sides to get: \( 2x = \frac{5\pi}{6} + \frac{\pi}{3} + 2k\pi \).Combine fractions: \( 2x = \frac{5\pi}{6} + \frac{2\pi}{6} + 2k\pi = \frac{7\pi}{6} + 2k\pi \).Divide both sides by 2: \( x = \frac{7\pi}{12} + k\pi \).

Step 4: Combine the Solutions

Combine both sets of solutions: \( x = \frac{\pi}{4} + k\pi \) and \( x = \frac{7\pi}{12} + k\pi \), where \( k \) is any integer.

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \theta \), is a fundamental trigonometric function that maps any angle to a value between -1 and 1. This function helps describe wave-like patterns and oscillations.

• It is periodic with a period of \( 2\pi \), meaning that its values repeat after every \( 2\pi \) radians or 360 degrees.
• The sine function reaches its maximum value of 1 and minimum value of -1.
• The function is symmetric about the origin, making it an odd function with the property \( \sin(-\theta) = -\sin(\theta) \).

Understanding the sine function is crucial for solving trigonometric equations because it directly relates angles and ratios in right triangles and unit circle calculations.

Equation Solving

Equation solving in trigonometry involves finding all possible angle measures that satisfy a given trigonometric equation. In this exercise, we aim to solve \( \sin(2x - \frac{\pi}{3}) = \frac{1}{2} \). Here are the steps to solve such equations:

• Identify Known Values: First, you determine at which angles the sine function equals the given value by solving \( \sin(\theta) = \frac{1}{2} \).
• Set Up Equation: Equate the inner expression \( 2x - \frac{\pi}{3} \) to the angles where the sine function takes on the value \( \frac{1}{2} \).
• Adjust for Periodicity: Because the sine function is periodic, the solution for \( \theta \) will include all angles that are equivalent to \( \frac{\pi}{6} \) or \( \frac{5\pi}{6} \) plus any multiple of \( 2\pi \).
• Solve for \( x \): This generally involves isolating \( x \) for all possible integer values of \( k \).

This systematic approach is essential for solving more complex trigonometric equations effectively.

Angle Measures

Angle measures are units used to describe the size of angles. The most common units are degrees and radians.

• Degrees: A full circle is divided into 360 degrees. This is the more intuitive system for everyday measurements.
• Radians: Based on the radius of the circle, it's a more natural measurement for mathematical applications. A full circle is \( 2\pi \) radians. Conversion between radians and degrees is essential, with the formula: \( \, 180^{\circ} = \pi \, \text{radians} \).

Understanding how to navigate between these units is crucial for solving trigonometric equations because trigonometric functions in calculus are typically expressed in radians. The solution to the equation is displayed using angle measures in radians, which are more practical for most high-level math applications.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is an essential tool in trigonometry for visually understanding the relationships between angles and trigonometric functions.

• Circle Basics: Because the radius is 1, each point on the circle can be described by the coordinates \( (\cos \theta, \sin \theta) \).
• Key Angles: The unit circle allows easy access to the sine and cosine values of commonly used angles such as \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \).
• Periodicity and Symmetry: The circle showcases the periodic nature of trigonometric functions, and symmetries can be used to find equivalent angles across different quadrants.

Utilizing the unit circle simplifies the understanding of trigonometric functions, making the process of solving equations like the one from the exercise clearer and more intuitive.