Questions: Solve for x in 4 cos ^2(x)=1 x=-60^circ x=60^circ x=120^circ x=-60^circ, 60^circ x=60^circ, 120^circ

Transcript text: Solve for $x$ in $4 \cos ^{2}(x)=1$ $x=-60^{\circ}$ $x=60^{\circ}$ $x=120^{\circ}$ $x=-60^{\circ}, 60^{\circ}$ $x=60^{\circ}, 120^{\circ}$

Solution

Solution Steps

Step 1: Rewrite the equation

Start with the equation $ 4 \cos^{2}(x) = 1 $. Divide both sides by 4 to isolate $ \cos^{2}(x) $: \[ \cos^{2}(x) = \frac{1}{4}. \]

Step 2: Solve for $ \cos(x) $

Take the square root of both sides to solve for $ \cos(x) $: \[ \cos(x) = \pm \frac{1}{2}. \]

Step 3: Find the values of $ x $

Determine the angles $ x $ where $ \cos(x) = \frac{1}{2} $ and $ \cos(x) = -\frac{1}{2} $:

For $ \cos(x) = \frac{1}{2} $, the solutions are $ x = 60^{\circ} $ and $ x = -60^{\circ} $.
For $ \cos(x) = -\frac{1}{2} $, the solutions are $ x = 120^{\circ} $ and $ x = -120^{\circ} $.

Thus, the possible solutions are $ x = -60^{\circ}, 60^{\circ}, 120^{\circ}, -120^{\circ} $.