Problem 77 - Precalculus with Limits

Question

In Exercises 75 - 78, use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. \( 4 \cos^2 x - 2 \sin x + 1 = 0 \), \( \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right] \)

Step-by-Step Solution

Answer

This is an approximation task and the answer depends on the graph obtained. The solutions will be read off from the intersection point of graph with the x-axis, within the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\).

Step 1: Rewrite the equation

Rewrite the given equation in terms of cosine function only. Use the Pythagorean Identity \(\sin ^2 x = 1- \cos ^2x \) to rewrite \(\sin x\) as \(\sqrt{1 - \cos^2 x} \). So, the equation becomes \(4\cos^2 x - 2\sqrt{1 - \cos^2 x} + 1 = 0\)

Step 2: Graph the equation

Use a graphing utility to graph the equation \(4\cos^2 x - 2\sqrt{1 - \cos^2 x} + 1 \). Make sure to graph the function on the given interval \(-\frac{\pi}{2}, \frac{\pi}{2}\). Also, plot the line \( y =0 \) for easy visualization of where the function intersects the x-axis

Step 3: Identify intersections

Identify the points of intersection of the function graph with the x-axis. These points represent the solutions of the equation.

Step 4: Read-off x-values

Based on the plotted graph, read off the x-values of the points of intersection. Round your answers to three decimal places. These coordinates will be the solutions to the given equations within the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship between the trigonometric functions sine and cosine. It states that for any angle \( x \), the square of the sine of \( x \), plus the square of the cosine of \( x \), is equal to one. This can be written as the equation \( \text{sin}^2x + \text{cos}^2x = 1 \). This identity is extremely useful when solving trigonometric equations because it allows you to convert between sine and cosine.

In the given exercise, the Pythagorean Identity is used to transform the original equation so that it contains only one trigonometric function, cosine in this case. This simplification is a common technique for solving trigonometric equations because it reduces the complexity of the problem, making it easier to graph and solve.

Graphing Utility

A graphing utility is an essential tool for visualizing and solving trigonometric equations. It can be a software program, an online graphing calculator, or a graphing calculator device. These utilities allow you to plot the behavior of trigonometric functions over a selected interval, which is critical for identifying the points where the function meets certain conditions—like crossing the x-axis in the case of finding roots.

When solving the given equation, the use of a graphing utility enables you to see the points where the graph intersects the x-axis, which represent the solutions. It's important to set the interval correctly; for the exercise, it was on \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). This aids in approximating the solutions to the required degree of precision which, in this case, is three decimal places.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are relationships that provide a link between an angle and a ratio of sides in a right triangle. They are also defined as functions on the unit circle, which is the foundation for their use in various applications including solving equations, modeling periodic phenomena, and in many areas of science and engineering.

The cosine function, which is used in the provided exercise, measures the x-coordinate of a point on the unit circle corresponding to a given angle. Cosine functions, like all trigonometric functions, are periodic and exhibit a wave-like pattern—understanding this behavior is key to predicting the solutions to trigonometric equations and graphing them effectively.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy the equation. The general approach often includes simplifying the equation using trigonometric identities, isolating the trigonometric function, and then using inverse trigonometric functions to find the angle solutions. However, when an equation is more complex or does not lend itself easily to analytical solutions, graphing is an extremely helpful method.

As demonstrated in the exercise, the graphing method allows visual identification of the solutions. Once you graph the function and the line \( y=0 \), the x-values where the two intersect are the solutions within the selected interval. It's crucial to be precise in rounding to the required number of decimal places to ensure the correct answers. Understanding the periodic nature of trigonometric functions also helps in anticipating the number of solutions one might expect within a given interval.