91Ó°ÊÓ

In trigonometry, double angle identities are incredibly useful as they allow us to transform and simplify expressions. For this problem, the double angle identity for cosine is particularly important. It is expressed as \( \cos 2x = 2\cos^2 x - 1 \). This identity helps to relate the cosine of a double angle \( 2x \) to the square of a single angle cosine \( \cos x \).Why use double angle identities?

• They simplify complex trigonometric expressions, making equations easier to solve.
• They can be used to transform equations into a form that is more manageable, often leading to quadratic equations.
• Helps in finding specific angle measures by resolving them back to their respective identities.

In this exercise, applying the double angle identity transforms the expression \( \cos 2x \) into \( 2\cos^2 x - 1 \), allowing further simplification and leading to a quadratic equation. This transformation is a crucial step for solving the trigonometric equation presented.

Once the double angle identity has been applied, the original equation becomes a quadratic equation in terms of cosine, such as \( 2\cos^2 x - 2\cos x - 1 = 0 \). Solving quadratic equations is a fundamental skill in trigonometry when these forms emerge.Quadratic equations generally take the form \( ax^2 + bx + c = 0 \), and can be solved using methods like:

• Factoring, when expressions can smoothly be broken into binomials.
• The quadratic formula, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides solutions even when factoring is difficult.
• Graphical analysis, where you can find solutions by identifying where the parabola crosses the x-axis.

In our exercise, solving the quadratic equation gives possible values for \( \cos x \), specifically \( \cos x = -0.5 \) and \( \cos x = 1 \). Solving these values leads directly to determining the possible angles \( x \) on the unit circle.

Solving the quadratic equation in the context of this problem yields the cosine values \( \cos x = -0.5 \) and \( \cos x = 1 \). It's crucial to remember how to interpret these cosine values to find angle solutions over specific intervals.Identifying angles:

• For \( \cos x = 1 \), cosine equals 1 only at \( x = 0^{\circ} \) and \( 360^{\circ} \).
• For \( \cos x = -0.5 \), refer to a unit circle or trigonometric table. The solutions are at angles where cosine is negative half, specifically \( x = 120^{\circ} \) and \( x = 240^{\circ} \).

To solve these, it is important to account for the periodicity of the cosine function, which repeats every \( 360^{\circ} \). Therefore, angles that satisfy \( \cos x = -0.5 \) and fall within \( 0^{\circ} \) to \( 360^{\circ} \) are required. Combining this information narrows down solutions to specific options in answer choices. This process confirmed the solution range; however, a mismatch in proposed question options suggests checking calculations or the potential for inaccuracies in the problem's options given.