Problem 149 - Precalculus Essentials

Question

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$2 \sin ^{2} x=1-2 \sin x$$

Step-by-Step Solution

Answer

The exact solutions will depend on how accurately the intersection points were detected. However, all the solutions will be a value in the interval [0,2π), rounded to the nearest hundredth.

Step 1: Graph the Function

Firstly, one should graph the two functions \(2 \sin ^{2} x\) and \(1-2 \sin x\) individually on the graphing calculator. Make sure that the graphing mode is set to radians and the range for the X-axis is limited to the interval [0,2π).

Step 2: Find the Intersection Points

After graphing the functions, the intersection points of the two graphs are the solutions of the equation. Therefore, use the 'intersect' feature of your graphing tool to find the X-values of the intersection points. Remember that the 'intersect' command asks for a guess. Iterate this process until no new intersection points are found in the range [0,2π).

Step 3: Round to the nearest hundredth

Finally, round all the obtained X-values to the nearest hundredth. These are the solutions to the equation \(2 \sin ^{2} x = 1-2 \sin x\) in the interval [0,2π).

Key Concepts

Graphing Utility

To solve trigonometric equations effectively, a graphing utility, like a graphing calculator or a software tool, can be quite helpful. It visualizes expressions and functions, enabling you to spot key features like intersections, maximums, or zeros. When solving equations, ensure that your graphing utility is set correctly to analyze the specific interval required.

For this exercise, your calculator should be set to:

• Graphing mode set to radians, which is essential for trigonometric functions.
• X-axis limited to the interval \( [0, 2\pi) \), ensuring you only find solutions within the necessary range.

Taking time to set it up right can save many missteps later. By graphing both functions, you'll better visualize potential intersections, which are key to finding solutions.

Intersection Points

In graphing, intersection points are where two or more functions cross each other. These points represent solutions to the equation formed by setting two functions equal to each other. In this exercise, the graphing utility helps you find where the graph of \(2 \sin^2 x\) intersects with the graph of \(1 - 2 \sin x\).

To pinpoint these intersections:

• Use the 'intersect' function provided by the graphing tool.
• This function may ask you for an initial guess; start near where the graphs seem to touch.
• Iterate further to ensure no intersection is missed within the interval \( [0, 2\pi) \).

Intersection points within the specified range provide the solution to the equation in the trigonometric context.

Radians

Radians are a unit of angular measure used extensively in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians relate angles to \(\pi\), capturing the relationship between an angle's arc length and the circle's radius. This makes calculations and interpretations more straightforward in advanced mathematics.

For this problem:

• The interval \( [0, 2\pi) \) represents a full circle (0 to 360 degrees) but measured in radians.
• All solutions should be expressed in radians, rounded to the nearest hundredth.

Working in radians ensures consistency and accuracy, especially when using graphing utilities for trigonometric functions.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental to understanding periodic phenomena. In this exercise, you work with the sine function, which oscillates between -1 and 1.

• \( \sin x \) is the trigonometric function representing the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• \( 2 \sin^2 x \) is a transformation of the basic sine function. It stretches and shifts the sine waves in specific ways.
• \( 1 - 2 \sin x \) is another transformation involving subtraction from a constant, which affects its graph and how it intersects with other functions.

By understanding how these transformations work, you can more easily predict and interpret intersections, which are crucial for finding solutions.