Solving the Equation sin(x) 1/2: A Comprehensive Guide

In this article, we will walk you through the process of solving the equation sin(x) 1/2. By the end, you will understand how to identify reference angles, determine general solutions, and solve related trigonometric equations. Whether you are a student, a teacher, or someone interested in mathematics, this guide will provide you with the necessary tools to solve such equations accurately.

Identifying the Reference Angle

The first step in solving the equation sin(x) 1/2 is to identify the reference angle. The sine function equals 1/2 at specific angles. The reference angle where sin(x) 1/2 is x π/6 or 30°. This angle is crucial as it helps us determine the primary solutions in the unit circle.

Determining the General Solutions

To find the general solutions, we need to consider the periodic nature of the sine function and its behavior in different quadrants:

First Quadrant

In the first quadrant, the sine function is positive. Therefore, one solution is:

x π/6 2kπ

Second Quadrant

In the second quadrant, the sine function is still positive. Another solution is:

x (π - π/6) 2kπ 5π/6 2kπ

Combining the Solutions

The complete solution set, considering the periodicity of the sine function, is:

x π/6 2kπ and x 5π/6 2kπ

where k is any integer. This means that the solutions repeat every 2π, covering all possible angles where sin(x) 1/2.

Using the Inverse Sine Function

Another method to find the solutions is to use the inverse sine function, denoted as arcsin. Applying the inverse sine to both sides of the equation:

sin(x) 1/2

gives:

x arcsin(1/2)

The value of arcsin(1/2) is π/6 or 30°. However, as mentioned earlier, the sine function is positive in both the first and second quadrants, thus:

x π/6 or 5π/6

Alternative Method: Solving 2x 1/2

For the equation 2x 1/2, the solution can be found by dividing both sides by 2:

x 1/4

However, this method is not directly related to the equation sin(x) 1/2. It is a different type of equation and should be solved accordingly.

General Solution Summary

The complete set of solutions for sin(x) 1/2 is:

x π/6 2kπ and x 5π/6 2kπ

where k is any integer.

Understanding the Sine Wave Function

The sine wave function sin(x) continuously varies from 0 up to 1, down to -1, and back to 0 in a cycle of the abscissa, which varies from 0 to 2π (a full circle). The function sin(x) hits every value between -1 and 1. At a specific value of the abscissa θ, the sine function hits 1/2. This value is particularly sin(30°) 1/2.

Common Applications and Practice

To familiarize yourself with solving trigonometric equations, you can practice by substituting values for x until you are close enough to the solution. For more complex equations, understanding the periodic nature and reference angles is essential. Regular practice and using resources like trigonometric tables and calculators can enhance your problem-solving skills.

Conclusion

By mastering the techniques to solve the equation sin(x) 1/2, you can tackle a wide range of trigonometric problems. This guide provides a step-by-step approach, helping you to identify reference angles, determine general solutions, and apply inverse functions. Regular practice and a clear understanding of the sine wave function will enable you to solve more complex trigonometric equations with confidence.