$ ext{Find the Value of Cosine on the Unit Circle}$

$\text{Given the unit circle, we need to find the value of } \cos(\theta) \text{ where } \theta \text{ is an angle such that } 2\cos^2(\theta) + \cos(\theta) – 1 = 0.$

$\text{Step 1: Solve the quadratic equation} $

$2\cos^2(\theta) + \cos(\theta) – 1 = 0$

$\text{Using the quadratic formula } x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}, \text{ where } a = 2, b = 1, \text{ and } c = -1$

$\cos(\theta) = \frac{-1 \pm \sqrt{1^2 – 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4}$

$\text{Thus, } \cos(\theta) = \frac{2}{4} = \frac{1}{2} \text{ or } \cos(\theta) = \frac{-4}{4} = -1.$

$\text{Therefore, the possible values of } \cos(\theta) \text{ are } \boxed{\frac{1}{2} \text{ and } -1}.$

$ ext{Consider the unit circle and the equation } 2cos^2( heta) + cos( heta) – 1 = 0.$

$ ext{Step 1: Factor the quadratic expression} $

$2cos^2( heta) + cos( heta) – 1 = 0$

$ ext{Rewriting the equation, we get} $

$(2cos( heta) – 1)(cos( heta) + 1) = 0$

$ ext{Setting each factor to zero, we find} $

$2cos( heta) – 1 = 0 Rightarrow cos( heta) = frac{1}{2}$

$cos( heta) + 1 = 0 Rightarrow cos( heta) = -1$

$ ext{Therefore, } cos( heta) = oxed{frac{1}{2} ext{ and } -1}.$

$ ext{Solve } 2cos^2( heta) + cos( heta) – 1 = 0 ext{ to find } cos( heta).$

$ ext{Using the quadratic formula:}$

$cos( heta) = frac{-1 pm sqrt{1 + 8}}{4} = oxed{frac{1}{2} ext{ and } -1}.$

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