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: Evaluate each trigonometric term using special angles and quadrant rules. For 135^: 135^ is in Quadrant II. The reference angle is 180^ - 135^ = 45^. In Quadrant II, sine is positive. 135^ = 45^ = sqrt(2)2 For 360^: 360^ is equivalent to 0^. 360^ = 0^ = 1 For 150^: 150^ is in Quadrant II. The reference angle is 180^ - 150^ = 30^. In Quadrant II, sine is positive. 150^ = 30^ = (1)/(2) For 225^: 225^ is in Quadrant III. The reference angle is 225^ - 180^ = 45^. In Quadrant III, cosine is negative. 225^ = - 45^ = -sqrt(2)2 Step 2: Substitute these values into the given expression. The expression is: ( 135^ - 360^)/( 150^ + 225^) Substitute the values: sqrt(2)2 - 1(1)/(2) + (-sqrt(2)2) sqrt(2)2 - 1(1)/(2) - sqrt(2)2 Step 3: Simplify the numerator and the denominator. Numerator: sqrt(2)2 - 1 = sqrt(2) - 22 Denominator: (1)/(2) - sqrt(2)2 = 1 - sqrt(2)2 Step 4: Perform the division. sqrt(2) - 221 - sqrt(2)2 = sqrt(2) - 22 × (2)/(1 - sqrt(2)) = sqrt(2) - 21 - sqrt(2) Step 5: Rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is 1 + sqrt(2). (sqrt(2) - 2)(1 + sqrt(2))(1 - sqrt(2))(1 + sqrt(2)) Expand the numerator: (sqrt(2) - 2)(1 + sqrt(2)) = sqrt(2)(1) + sqrt(2)(sqrt(2)) - 2(1) - 2(sqrt(2)) = sqrt(2) + 2 - 2 - 2sqrt(2) = -sqrt(2) Expand the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2: (1 - sqrt(2))(1 + sqrt(2)) = 1^2 - (sqrt(2))^2 = 1 - 2 = -1 Now, substitute these back into the fraction: -sqrt(2)-1 = sqrt(2) The simplified expression is sqrt(2). 3 done, 2 left today. You're making progress.