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The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It's primarily used to determine the horizontal coordinate of a point on the unit circle. Understanding cosine is crucial as it helps solve various trigonometric equations and problems. For an angle \( \theta \), \( \cos \theta \) gives the x-coordinate of the corresponding point on the unit circle.

Key properties of the cosine function include:

• Range: The values of cosine range from -1 to 1.
• Periodicity: The cosine function is periodic with a period of \( 2\pi \). This means that \( \cos(\theta) = \cos(\theta + 2\pi n) \) for any integer \( n \).
• Symmetry: The function is even, meaning \( \cos(-\theta) = \cos(\theta) \).

These characteristics help in identifying solutions within specific intervals, such as between 0 and \( 2\pi \), as seen in the given problem. By knowing these angles, we can identify particular solutions that meet the equation’s criteria.

In trigonometry, quadratic equations frequently appear, often hidden within trigonometric functions like cosine. A quadratic equation is generally in the form \( ax^2 + bx + c = 0 \). Solving them can yield the specific values needed for additional calculations.

To solve a quadratic equation, there are several methods, with the quadratic formula being among the most reliable:

• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Factoring: If easily factorable, you can rewrite the equation to find its roots directly.
• Completing the Square: A method where the equation is rearranged into a perfect square trinomial.

For cosine-related equations, substitution is often used to transform a trigonometric expression into a quadratic equation, making it easier to solve. In the exercise, \( \cos \theta \) was substituted with \( x \) to apply the quadratic formula, solving for the cosine values that satisfy the original equation.

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. These identities simplify and solve complex trigonometric equations. Knowing and applying these can vastly simplify problem-solving.

Here are critical identities relevant to understanding trigonometric solutions:

• Double Angle Identity: \( \cos 2\theta = 2\cos^2 \theta - 1 \) helps rewrite terms involving \( \cos 2\theta \) into useful quadratic expressions.
• Pythagorean Identity: \( \cos^2 \theta + \sin^2 \theta = 1 \). This relates the cosine and sine functions.

Using these identities, like the double angle identity in this exercise, turns challenging expressions into manageable quadratic equations. Consequently, it becomes straightforward to apply algebraic techniques for solutions. Recognizing when and how to use these identities is a fundamental skill in trigonometry.