Solve 2 (2x + 30) = 3 (2x + 30) for (2x)

Step 1: Simplify the given equation to find (2x+30). We are given the equation: 2 (2x + 30) = 3 (2x + 30) Divide both sides by (2x + 30): (2 (2x + 30))/((2x + 30)) = (3 (2x + 30))/((2x + 30)) 2 (2x + 30) = 3 (2x + 30) = (3)/(2) Step 2: Use trigonometric identities to express (90 - 2x) in terms of (2x). We know the co-function identity: (90^ - ) = () Therefore, for = 2x: (90^ - 2x) = (2x) Our goal is to find the value of (2x). Step 3: Use the tangent value to find (2x+30) and (2x+30). Let = 2x + 30. We have () = (3)/(2). We can construct a right-angled triangle where the opposite side to angle is 3 and the adjacent side is 2. Using the Pythagorean theorem, the hypotenuse h is: h = sqrt(3^2 + 2^2) = sqrt(9 + 4) = sqrt(13) From this triangle, we can find () and (): () = oppositehypotenuse = (3)/(sqrt(13)) () = adjacenthypotenuse = (2)/(sqrt(13)) Step 4: Express (2x) using the angle subtraction formula. We know that 2x = - 30^. Using the cosine subtraction formula (A - B) = A B + A B: (2x) = ( - 30^) = () (30^) + () (30^) We know the exact values for (30^) and (30^): (30^) = sqrt(3)2 (30^) = (1)/(2) Substitute the values of (), (), (30^), and (30^): (2x) = ((2)/(sqrt(13))) (sqrt(3)2) + ((3)/(sqrt(13))) ((1)/(2)) (2x) = 2sqrt(3)2sqrt(13) + (3)/(2sqrt(13)) (2x) = 2sqrt(3) + 32sqrt(13) Step 5: Calculate the numerical value and round to 4 decimal places. (2x) = 2sqrt(3) + 32sqrt(13) Using approximate values: sqrt(3) ≈ 1.7320508 sqrt(13) ≈ 3.6055513 (2x) ≈ (2(1.7320508) + 3)/(2(3.6055513)) (2x) ≈ (3.4641016 + 3)/(7.2111026) (2x) ≈ (6.4641016)/(7.2111026) (2x) ≈ 0.89641009 Rounding to 4 decimal places: (2x) ≈ 0.8964 Since (90^ - 2x) = (2x), (90^ - 2x) ≈ 0.8964 The final answer is 0.8964. That's 2 down. 3 left today — send the next one.