Problem 125 - Precalculus Essentials

Question

Solve each equation on the interval \([0,2 \pi)\) \(2 \cos ^{3} x+\cos ^{2} x-2 \cos x-1=0\) (Hint: Use factoring by grouping.)

Step-by-Step Solution

Answer

The solutions to the given equation within the interval \([0, 2\pi]\) are x = 0, \(\frac{2\pi}{3}\), \(\frac{4\pi}{3}\), and \(2\pi\).

Step 1: Rewrite the equation

First, notice that the given equation is similar to a quadratic equation if we treat \(cos x\) as 'x'. Thus, we rewrite the equation as \(2(cosx)^3 + (cosx)^2 - 2cosx - 1 = 0\), which resembles the structure of a trinomial quadratic equation.

Step 2: Factor by Grouping

Group together the terms so that we can factor separately. Group the first two terms and the last two terms: \(2(cosx)^3 + (cosx)^2 - (2cosx + 1) = 0\). Now, factor out common factors from each binomial: \(cosx((2cosx)^2 + 1) - 1(2cosx + 1) = 0\). Set this equal to zero and factor out the common binomial, resulting in \((cosx - 1)(2cosx + 1) = 0\).

Step 3: Solve for Cosine Values

Now, set each factor equal to zero and solve. When \(cosx - 1 = 0\), \(cosx = 1\). When \(2cosx + 1 = 0\), \(cosx = -0.5\). This gives us the possible cosine values within the given interval.

Step 4: Find Corresponding Angles

We know that in the interval \([0,2\pi]\), cosine equals 1 at x=0 and x=\(2\pi\), and cosine equals -0.5 at x=\(\frac{2\pi}{3}\), x=\(\frac{4\pi}{3}\). Thus we find the solution to our problem.

Key Concepts

Cosine Function

The cosine function is one of the primary functions in trigonometry. It is often used to describe the relationship between the angle and the ratio of the adjacent side to the hypotenuse in a right triangle.

The function is denoted as \( \cos(x) \) and it provides a measure of the projection of a point on a unit circle onto the horizontal axis, as that point moves around the circle.

• The cosine function is periodic with a period of \( 2\pi \).
• Its range is from -1 to 1.
• Key positions include 0, \( \pi \), and \( 2\pi \), where cosine equals 1, -1, and again 1, respectively.

Understanding how \( \cos(x) \) behaves over its domain helps us to solve trigonometric equations and find solutions within specific intervals.

Trigonometric Equations

Trigonometric equations involve trigonometric functions, like sine or cosine, and are solved to find the angles that make the equation true. In trigonometric equations, recognizing patterns or identities can simplify solving.

To solve equations involving the cosine function, such as \( 2 \cos^3 x + \cos^2 x - 2 \cos x - 1 = 0 \), factoring techniques are often deployed to find feasible solutions.

• If an equation can be rewritten as a factored equation, like \((\cos x - 1)(2 \cos x + 1)=0\), you simplify finding the roots.
• Each set to zero gives potential values for cosine at particular angles.

By understanding the trigonometric properties and manipulating the equation via algebraic methods, you can solve for angles that satisfy the equation, ensuring solutions fall within any given interval boundaries.

Interval Notation

Interval notation is a system used to express the set of solutions or domain of variables. It’s particularly useful when showcasing the range of angles for trigonometric equations.

Interval notation uses brackets "[]" and parentheses "()" to indicate whether endpoints are included or not.

• \([0, 2\pi)\) indicates that 0 is included in the interval, while \(2\pi\) is not.
• This notation is essential when solving problems in trigonometry to ensure all solutions are found within specified constraints.
• Utilizing intervals ensures clear communication of solution sets, especially in contexts where periodic functions like cosine are involved.

Using interval notation helps ensure you respect the boundaries of a problem, especially in cases where the angle's period impacts multiple cycles.