91Ó°ÊÓ

Quadratic equations are equations of the second degree, generally written as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). In the context of trigonometric equations, these constants often involve trigonometric functions like sine or cosine.
In the given exercise, we encounter a quadratic equation in terms of the cosine function: \(\cos^2 x - \cos x - 1 = 0\). This can be solved using the quadratic formula:

• Identify the coefficients: \(a = 1\), \(b = -1\), and \(c = -1\).
• Plug them into the formula: \(\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Simplify to find \(\cos x = \frac{1 \pm \sqrt{5}}{2}\).

This process helps us determine the possible values for \(\cos x\), which are helpful in finding the angles \(x\) within the specified intervals. However, not all solutions for \(\cos x\) are valid, as the equation must respect the range of cosine values, specifically between -1 and 1.

The cosine function is one of the fundamental trigonometric functions. It relates the adjacent side to the hypotenuse in a right-angled triangle, defined as the ratio \(\cos x = \frac{adjacent}{hypotenuse}\).
Cosine has several important properties:

• Its range is between -1 and 1. This means any equation involving \(\cos x\) will only have valid solutions if \(\cos x\) falls within this range.
• It is periodic with a period of \(2\pi\), meaning \(\cos(x + 2\pi) = \cos x\).

In the exercise, we find that \(\cos x\) can potentially be -2. However, because the cosine function's range does not permit values less than -1 or greater than 1, this must be discarded as an invalid solution. By understanding the function's constraints, we are guided to valid solutions, such as \(\cos x = 1\), which corresponds to specific angles in the given interval [0, \(2\pi)\).

Interval notation is a mathematical notation used to describe a set of numbers between two endpoints. In the context of solving equations, it helps specify which solutions are valid based on a given range.
In the exercise, we are asked to find solutions in the interval \([0, 2\pi)\). This means:

• The solutions should be bounded between 0 and \(2\pi\).
• It includes 0 (denoted by the bracket \([\)), making it part of the solution set.
• It excludes \(2\pi\) (denoted by the parenthesis \(()\)), indicating it is not included in the solution set.

Understanding interval notation is crucial when limiting the solutions of trigonometric equations. It ensures that the solutions meet the criteria set by the problem, such as keeping angles within the specified range, ensuring that our results are valid within the mathematical context provided.