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
B. 30^, 60^ Step 1: Find the principal values for 2x. Given the equation 2x = sqrt(3)2. The reference angle for = sqrt(3)2 is 60^. Since 2x is positive, 2x must be in Quadrant I or Quadrant II. The principal values for 2x are: 2x = 60^ or 2x = 180^ - 60^ = 120^ Step 2: Find the general solutions for 2x. The general solutions for = k are = + 360^ n or = (180^ - ) + 360^ n, where is the reference angle and n is an integer. So, for 2x: 2x = 60^ + 360^ n or 2x = 120^ + 360^ n Step 3: Solve for x. Divide both general solutions by 2: x = (60^)/(2) + (360^ n)/(2) x = 30^ + 180^ n or x = (120^)/(2) + (360^ n)/(2) x = 60^ + 180^ n Step 4: Apply the given range 0^ x 180^. For x = 30^ + 180^ n: • If n=0, x = 30^ + 180^(0) = 30^. This is within the range. • If n=1, x = 30^ + 180^(1) = 210^. This is outside the range. For x = 60^ + 180^ n: • If n=0, x = 60^ + 180^(0) = 60^. This is within the range. • If n=1, x = 60^ + 180^(1) = 240^. This is outside the range. The solutions for x in the given range are 30^ and 60^. The final answer is B. 30^, 60^.