Problem 97 - Precalculus: Functions and Graphs

Question

Estimate the solutions of the equation in the interval \([-\pi, \pi]\). $$3 \cos ^{4} x-2 \cos ^{3} x+\cos x-1=0$$

Step-by-Step Solution

Answer

The approximate solutions are \(-\pi, -\frac{\pi}{3}, \frac{\pi}{3}, \pi\) within \([-\pi, \pi]\).

Step 1: Simplify the Equation

Rewrite the equation as a polynomial in terms of \(y = \cos x\). This allows easier manipulation and estimation: \[3y^4 - 2y^3 + y - 1 = 0\] where \(-1 \leq y \leq 1\).

Step 2: Examine Interval Bounds

Identify reasonable bounds for \(y\) (where \(-1 \leq y \leq 1\)) given the trigonometric nature of the equation. Since \(\cos x\) maps to this interval, solutions must lie within this range.

Step 3: Apply Synthetic Division or Root Estimation Techniques

Use techniques like the Rational Root Theorem or synthetic division to estimate potential roots of the polynomial \(3y^4 - 2y^3 + y - 1 = 0\). Check values like \(-1, -0.5, 0, 0.5, 1\) to see if any satisfy the polynomial equation.

Step 4: Solve for \(x\) Using Estimated Roots

Assume estimated roots from Step 3 are directly usable for \(\cos x\). This transforms the problem into finding \(x\) such that \(\cos x = y\) where \(y\) is a root. Use inverse trigonometric functions to find \(x\). For instance, if \(y = 0.5\), then \(x = \pm \frac{\pi}{3}\) within the interval \([-\pi, \pi]\).

Step 5: Verify Solutions in the Given Interval

Plug the solutions back into the original trigonometric equation to ensure they satisfy it. Only include solutions \(x\) within the interval \([-\pi, \pi]\).

Key Concepts

Polynomial Transformation

In many trigonometric equations, simplifying the expression by transforming it into a polynomial can make solving the equation more approachable. This method is particularly useful when dealing with higher-degree trigonometric functions. For our problem, the equation \(3 \cos^{4} x - 2 \cos^{3} x + \cos x - 1 = 0\) can be rewritten by substituting \(y = \cos x\). This leads to a transformed polynomial:

• \(3y^4 - 2y^3 + y - 1 = 0\)

This change allows us to view our trigonometric problem as a fourth-degree polynomial problem, which behaves according to familiar algebraic rules. Polynomial transformations like this help to explore solutions more clearly, especially when dealing with multi-step problems or complex trigonometric forms.

Interval Estimation

Interval estimation is crucial when solving trigonometric equations, as it narrows down the scope of possible solutions. Specifically, for trigonometric functions like cosine, we know that its values are constrained between

• \(-1 \leq y \leq 1\)

This constraint indicates that any solution \(y = \cos x\) must lie within this range. The interval

• \([-\pi, \pi]\)

given in the exercise specifies where the angle \(x\) must be. By determining this interval, we limit both the inputs and outputs associated with our equation, making it easier to focus our solution efforts on plausible values of \(y\). Estimating intervals is essential for efficiently solving and checking solutions.

Synthetic Division

Synthetic division is a valuable tool for finding potential roots of polynomial equations. This method streamlines the process of division, making it quicker to identify whether certain values satisfy the polynomial equation. For the polynomial \(3y^4 - 2y^3 + y - 1 = 0\), using synthetic division can help test candidate roots such as

• \(-1, \-0.5, \0, \0.5, \1\)

The goal is to find roots that equate the polynomial to zero. Successfully finding such roots provides us with potential values of \(y\), which in turn represents possible cosine values of the original trigonometric equation. Synthetic division not only speeds up testing candidate roots but also reduces manual computation errors.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to convert the output of a trigonometric function back into the corresponding angle. In this exercise, once we ascertain values like \(y = \cos x\) from polynomial roots, we use inverse functions to solve for \(x\).

• If \(y = 0.5\), then solving \(\cos x = 0.5\) involves using the inverse cosine function:

• \(x = \cos^{-1}(0.5)\), which leads to \(x = \pm \frac{\pi}{3}\) in the designated interval \([-\pi, \pi]\).

Inverse trigonometric functions help transition from general values back to specific angles, making them pivotal in verifying solutions to trigonometric equations. By ensuring \(x\) is within the stipulated interval, the process confirms that the solution is both viable and valid in the context of the original equation.