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Magazine Chapter 7: Problem 17
Exer. 1-38: Find all solutions of the equation. $$ \sin \left(2 x-\frac{\pi}{3}\right)=\frac{1}{2} $$
Short Answer
Expert verified
The solutions are \( x = \frac{\pi}{4} + k\pi \) and \( x = \frac{7\pi}{12} + k\pi \), where \( k \) is an integer.
Step by step solution
01
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.
02
Solve for x using the First Solution for 胃
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: \( x = \frac{\pi}{4} + k\pi \)
03
Solve for x Using the Second Solution for 胃
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: \( x = \frac{7\pi}{12} + k\pi \).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( \sin(-\theta) = -\sin(\theta) \).
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 \( k \).
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: \( \, 180^{\circ} = \pi \, \text{radians} \).
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.
