This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Isolate 2x. Divide both sides of the equation by 4: 4 2x = -2 2x = (-2)/(4) 2x = -(1)/(2) Step 2: Determine the range for 2x. Given the range for x is 0^ ≤ x ≤ 360^. Multiply the inequality by 2 to find the range for 2x: 2 · 0^ ≤ 2x ≤ 2 · 360^ 0^ ≤ 2x ≤ 720^ Step 3: Find the principal values for 2x. The cosine function is negative in the second and third quadrants. The reference angle for which = (1)/(2) is 60^. In the second quadrant, 2x = 180^ - 60^ = 120^. In the third quadrant, 2x = 180^ + 60^ = 240^. Step 4: Find all solutions for 2x within the range 0^ ≤ 2x ≤ 720^. The general solutions for = -(1)/(2) are = 120^ + n · 360^ and = 240^ + n · 360^, where n is an integer. For 2x = 120^ + n · 360^: • If n=0, 2x = 120^. • If n=1, 2x = 120^ + 360^ = 480^. • If n=2, 2x = 120^ + 720^ = 840^ (This is outside the range 0^ ≤ 2x ≤ 720^). For 2x = 240^ + n · 360^: • If n=0, 2x = 240^. • If n=1, 2x = 240^ + 360^ = 600^. • If n=2, 2x = 240^ + 720^ = 960^ (This is outside the range 0^ ≤ 2x ≤ 720^). So, the values for 2x are 120^, 240^, 480^, 600^. Step 5: Solve for x. Divide each value of 2x by 2: • x = (120^)/(2) = 60^ • x = (240^)/(2) = 120^ • x = (480^)/(2) = 240^ • x = (600^)/(2) = 300^ The solutions for x in the given range are 60^, 120^, 240^, 300^. x = 60^, 120^, 240^, 300^ That's 2 down. 3 left today — send the next one.