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Solving the equation \(2\cos^2(x) - \cos(x) - 1 = 0\) involves recognizing it as a quadratic equation in terms of \(\cos(x)\). The general form of a quadratic equation is \(ax^2 + bx + c = 0\). In our case, we substitute \(u = \cos(x)\) to simplify the equation to \(2u^2 - u - 1 = 0\). This substitution helps us treat the equation as a standard quadratic equation, allowing us to utilize the quadratic formula.

• The quadratic formula: \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Here, \(a = 2\), \(b = -1\), and \(c = -1\).
• First, calculate the discriminant, \(b^2 - 4ac\), to determine the nature of the roots.

Understanding how to solve these forms of equations is crucial, as it forms the backbone of finding the trigonometric values in our problem.Using the quadratic formula reveals the two potential real solutions for \(u\). From there, these solutions guide us to the specific x-values that satisfy the original trigonometric equation.

Trigonometric identities are crucial for transforming and simplifying trigonometric equations. In our equation \(\cos(2x) = \cos(x)\), applying the identity for \(\cos(2x)\) is the first step to simplifying the equation. The identity used here is \(\cos(2x) = 2\cos^2(x) - 1\), which allows us to rewrite the equation in terms of a single trigonometric function, \(\cos(x)\).This transformation is essential because it turns a potentially complex equation into a form that can be more easily manipulated, such as a quadratic.

• Identities like \(\cos(2x) = 2\cos^2(x) - 1\) and \(\sin^2(x) + \cos^2(x) = 1\) can help simplify expressions and reveal hidden relationships.
• These transformations are not only valuable in solving trigonometric equations but also in calculus and other advanced mathematics areas.

By mastering these identities, you can solve a variety of problems that initially seem challenging. The ability to see and manipulate these relationships is a pivotal skill in mathematics.

Interval notation is a handy way to describe the set of solutions for equations, particularly in trigonometric contexts, where the solutions are often restricted to specific intervals. For the equation \(\cos(2x) = \cos(x)\), we restrict our solutions to the interval \([0, 2\pi)\), meaning that x should satisfy the equation within one full rotation of the unit circle.

• The bracket \([\) indicates that 0 is included in the interval, meaning \(x = 0\) is considered a valid solution.
• The parenthesis \()\) indicates that \(2\pi\) is not included in the interval, excluding it as a potential solution to avoid overlap as \(0\) and \(2\pi\) represent the same point on the unit circle.

Understanding interval notation is vital because it precisely defines the range of acceptable solutions. This clarity is particularly significant when interpreting trigonometric solutions, ensuring that each solution is strictly within the prescribed boundaries. Such precision is key in mathematics, especially when solutions are cyclic or periodic like those in trigonometry.