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
Okay, m, let's solve part (b) of question 1. To find all solutions to the equation (2) + 3 = 2 in the interval 0^ 360^: Step 1: Use the double angle identity for (2). We know that (2) = 1 - 2^2. Substitute this into the given equation. (1 - 2^2) + 3 = 2 Step 2: Rearrange the equation into a quadratic form. 1 - 2^2 + 3 - 2 = 0 -2^2 + 3 - 1 = 0 Multiply by -1 to make the leading coefficient positive: 2^2 - 3 + 1 = 0 Step 3: Solve the quadratic equation for . Let x = . The equation becomes 2x^2 - 3x + 1 = 0. Factor the quadratic equation: (2x - 1)(x - 1) = 0 This gives two possible values for x: 2x - 1 = 0 x = (1)/(2) x - 1 = 0 x = 1 So, we have = (1)/(2) or = 1. Step 4: Find the values of for = 1 in the interval 0^ 360^. The only angle in this interval for which = 1 is: = 90^ Step 5: Find the values of for = (1)/(2) in the interval 0^ 360^. The principal value (in the first quadrant) for which = (1)/(2) is: _1 = 30^ Since is also positive in the second quadrant, the other solution in the interval is: _2 = 180^ - 30^ = 150^ Step 6: List all solutions. The solutions for in the interval 0^ 360^ are 30^, 90^, and 150^. The solutions are 30^, 90^, 150^. That's 2 down. 3 left today — send the next one.