Solve the trigonometric equation 4\[2 x\] + 4 x = 5 for 0^ x 360^.

Step 1: Use the identity ^2 x + ^2 x = 1. From this identity, we can write ^2 x = 1 - ^2 x. Substitute this into the given equation: 4(1 - ^2 x) + 4 x = 5 Step 2: Expand and rearrange the equation into a quadratic form. 4 - 4^2 x + 4 x = 5 Move all terms to one side to set the equation to zero: -4^2 x + 4 x + 4 - 5 = 0 -4^2 x + 4 x - 1 = 0 Multiply by -1 to make the leading coefficient positive: 4^2 x - 4 x + 1 = 0 Step 3: Solve the quadratic equation for x. This is a perfect square trinomial, which can be factored as (2 x - 1)^2 = 0. Alternatively, let y = x. The equation becomes 4y^2 - 4y + 1 = 0. Using the quadratic formula y = -b ± sqrt(b^2 - 4ac)2a: y = -(-4) ± sqrt((-4)^2 - 4(4)(1))2(4) y = 4 ± sqrt(16 - 16)8 y = 4 ± sqrt(0)8 y = (4)/(8) y = (1)/(2) So, x = (1)/(2). Step 4: Find the values of x in the range 0^ x 360^. We need to find angles x for which x = (1)/(2). The reference angle for x = (1)/(2) is 60^. Since x is positive, x lies in the first and fourth quadrants. In the first quadrant: x = 60^ In the fourth quadrant: x = 360^ - 60^ = 300^ Both 60^ and 300^ are within the specified range 0^ x 360^. The solutions are 60^, 300^.