Solve for \( \theta \) if : \( 2 \cos ^{2} 2 \theta+\cos 2 \theta-1=0, \theta \in\left[-180^{\circ} ; 180^{\circ}\right] \)

To solve the equation \( 2 \cos^2 2\theta + \cos 2\theta - 1 = 0 \), we can start by letting \( x = \cos 2\theta \). This transforms our equation into a standard quadratic form: \[ 2x^2 + x - 1 = 0 \] Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c = -1 \): \[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \] \[ x = \frac{-1 \pm \sqrt{1 + 8}}{4} \] \[ x = \frac{-1 \pm \sqrt{9}}{4} \] \[ x = \frac{-1 \pm 3}{4} \] This gives us two possible solutions for \( x \): 1. \( x = \frac{2}{4} = \frac{1}{2} \) 2. \( x = \frac{-4}{4} = -1 \) Now we will solve for \( \theta \) using these values of \( x \). ### Case 1: \( \cos 2\theta = \frac{1}{2} \) The general solutions for \( \cos 2\theta = \frac{1}{2} \) are: \[ 2\theta = 60^\circ + 360^\circ k \quad \text{or} \quad 2\theta = -60^\circ + 360^\circ k \] Dividing by 2 gives: \[ \theta = 30^\circ + 180^\circ k \quad \text{or} \quad \theta = -30^\circ + 180^\circ k \] For \( k = 0 \): - \( \theta = 30^\circ \) - \( \theta = -30^\circ \) For \( k = 1 \): - \( \theta = 210^\circ \) (not in the interval) - \( \theta = 150^\circ \) (not in the interval) For \( k = -1 \): - \( \theta = -150^\circ \) (in the interval) - \( \theta = -210^\circ \) (not in the interval) Thus, from this case, we have: \[ \theta = 30^\circ, -30^\circ, -150^\circ \] ### Case 2: \( \cos 2\theta = -1 \) The general solution for \( \cos 2\theta = -1 \) is: \[ 2\theta = 180^\circ + 360^\circ k \] Dividing by 2 gives: \[ \theta = 90^\circ + 180^\circ k \] For \( k = 0 \): - \( \theta = 90^\circ \) (in the interval) For \( k = -1 \): - \( \theta = -90^\circ \) (in the interval) For \( k = 1 \): - \( \theta = 270^\circ \) (not in the interval) Thus, from this case, we have: \[ \theta = 90^\circ, -90^\circ \] ### Final Solutions Combining all the solutions from both cases, we have: \[ \theta = 30^\circ, -30^\circ, -150^\circ, 90^\circ, -90^\circ \] Thus, the final solutions for \( \theta \) in the interval \( [-180^\circ, 180^\circ] \) are: \[ \boxed{30^\circ, -30^\circ, -150^\circ, 90^\circ, -90^\circ} \]