If \(2\sin^{2} \theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find the value of \(\theta\).

Answer Details

We are given an equation in terms of trigonometric functions, and we are asked to find the value of one of the angles involved. One possible way to approach this problem is to use trigonometric identities to simplify the equation and then solve for the unknown angle. First, we can use the identity \(\sin^{2}\theta + \cos^{2}\theta = 1\) to rewrite the left-hand side of the equation as \(\sin^{2}\theta = 1 - \cos^{2}\theta\). Substituting this into the given equation, we get: \[2(1-\cos^{2}\theta) = 1 + \cos\theta\] Expanding and rearranging terms, we obtain a quadratic equation in terms of \(\cos\theta\): \[2\cos^{2}\theta + \cos\theta - 1 = 0\] We can solve this quadratic equation using the quadratic formula: \[\cos\theta = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\] where \(a=2\), \(b=1\), and \(c=-1\). Plugging in these values, we get: \[\cos\theta = \frac{-1 \pm \sqrt{1+8}}{4} = \frac{-1 \pm 3}{4}\] Therefore, the two possible values of \(\cos\theta\) are \(\frac{1}{2}\) and \(-1\). However, we are given that \(0° \leq \theta \leq 90°\), which means that \(\cos\theta \geq 0\) in this range. Therefore, we must choose \(\cos\theta = \frac{1}{2}\), which gives: \[\cos\theta = \frac{1}{2} \implies \theta = \cos^{-1}\frac{1}{2} \approx 60°\] Therefore, the answer is (B) 60°.