91Ó°ÊÓ

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent, and are solved within a specified interval. These equations often require us to find angles that satisfy them within a given domain. In this exercise, the goal is to find all solutions for the equation \(2 \sin^2(x) = 1 - \cos(x)\) where \(0^\circ \leq x < 360^\circ\).
Understanding trigonometric identities is crucial, as they allow us to transform an equation into a more workable form. For instance, using the identity \(\sin^2(x) = 1 - \cos^2(x)\), we can substitute and rearrange terms to set up a solvable equation in terms of one trigonometric function.
Solving trigonometric equations often involves converting them into algebraic forms. Once in this form, common algebraic techniques, like factoring or using the quadratic formula, can be utilized. Each root found in the algebraic solution corresponds to particular angle(s) that solve the original trigonometric equation.

Quadratic equations are polynomial equations of degree two, commonly written in the form \(ax^2 + bx + c = 0\). They play a pivotal role in many mathematical problems, including trigonometric equations, as shown in this exercise.
When faced with a quadratic equation like \(2 \cos^2(x) - \cos(x) - 1 = 0\), the goal is to find the values of \(\cos(x)\) that satisfy it. One method to solve quadratic equations is the quadratic formula:

• Determine the coefficients \(a\), \(b\), and \(c\).
• Use the formula \(\cos(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Plugging in \(a = 2\), \(b = -1\), and \(c = -1\) yields the solutions for \(\cos(x)\). These solutions are essential because they bridge the gap between a polynomial form and trigonometric angles that solve the original equation.

The cosine function, denoted as \(\cos(x)\), is one of the primary trigonometric functions. It provides the measure of the adjacent side over the hypotenuse in a right triangle and defines smooth oscillations between -1 and 1.
In the unit circle framework, \(\cos(x)\) corresponds to the x-coordinate of a point on the circle, meaning it provides a convenient geometric interpretation of angles and their associated values.
For example, when we determine that \(\cos(x) = 1\), the angle is \(x = 0^\circ\). This is because at this position, the point on the unit circle aligns with the positive x-axis. Similarly, if \(\cos(x) = -\frac{1}{2}\), it occurs at \(x = 120^\circ\) and \(x = 240^\circ\). Knowing cosine values is instrumental in trigonometry because it helps convert between angles and trigonometric values, integral for solving trigonometric equations.