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 3. Given the equation 2 3 - 1 = 0. Add 1 to both sides: 2 3 = 1 Divide by 2: 3 = (1)/(2) Step 2: Find the principal value for 3. Let be the principal value such that = (1)/(2). = ((1)/(2)) = 60^ Step 3: Find all possible values for 3 within the given range. The range for is 0^ 360^. Therefore, the range for 3 is 3 × 0^ 3 3 × 360^, which is 0^ 3 1080^. Since 3 is positive, 3 lies in the first and fourth quadrants. The general solutions for x = are x = 2n ± (in radians) or x = n · 360^ ± (in degrees), where n is an integer. For n=0: 3 = 60^ 3 = 360^ - 60^ = 300^ For n=1: 3 = 360^ + 60^ = 420^ 3 = 360^ + (360^ - 60^) = 360^ + 300^ = 660^ For n=2: 3 = 2 · 360^ + 60^ = 720^ + 60^ = 780^ 3 = 2 · 360^ + (360^ - 60^) = 720^ + 300^ = 1020^ All these values are within the range 0^ 3 1080^. Step 4: Divide each value of 3 by 3 to find . _1 = (60^)/(3) = 20^ _2 = (300^)/(3) = 100^ _3 = (420^)/(3) = 140^ _4 = (660^)/(3) = 220^ _5 = (780^)/(3) = 260^ _6 = (1020^)/(3) = 340^ The values of are 20^, 100^, 140^, 220^, 260^, 340^. That's 2 down. 3 left today — send the next one.