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: Rewrite the trigonometric functions in terms of and . The given equation is 3( - ) = 2. We know that = (1)/() and = ()/(). Substitute these into the equation: 3((1)/() - ()/()) = 2 Step 2: Combine the terms inside the parenthesis. 3((1 - )/()) = 2 Step 3: Multiply both sides by to eliminate the denominator. Note that this implies ≠ 0. 3(1 - ) = 2^2 Step 4: Use the Pythagorean identity ^2 = 1 - ^2 to express the equation entirely in terms of . 3(1 - ) = 2(1 - ^2) Step 5: Expand and rearrange the equation into a quadratic form. 3 - 3 = 2 - 2^2 2^2 - 3 + 3 - 2 = 0 2^2 - 3 + 1 = 0 Step 6: Solve the quadratic equation for . Let x = . 2x^2 - 3x + 1 = 0 Factor the quadratic equation: (2x - 1)(x - 1) = 0 This gives two possible solutions for x: 2x - 1 = 0 x = (1)/(2) x - 1 = 0 x = 1 Step 7: Substitute back and find the general solutions for . Case 1: = 1 The general solution is = ()/(2) + 2n, where n is an integer. However, in Step 3, we established that ≠ 0. If = 1, then = ()/(2) + 2n, which means = 0. This would make and undefined in the original equation. Therefore, = 1 is an extraneous solution. Case 2: = (1)/(2) The general solutions are = ()/(6) + 2n and = (5)/(6) + 2n, where n is an integer. For these values of , ≠ 0, so the original terms and are well-defined. The solutions for are: = ()/(6) + 2n or = (5)/(6) + 2n, where n Z Send me the next one 📸