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
Alright agyenim — let's do this. To solve the trigonometric equation 2x - 5^2 x = (1)/(9), we need to express all terms using a single trigonometric function. We will use the double angle identity for 2x and the Pythagorean identity. Step 1: Rewrite the equation using trigonometric identities. We know that 2x = 1 - 2^2 x. Substitute this into the equation: (1 - 2^2 x) - 5^2 x = (1)/(9) Step 2: Simplify the equation. Combine the ^2 x terms: 1 - 7^2 x = (1)/(9) Isolate the ^2 x term: 7^2 x = 1 - (1)/(9) 7^2 x = (9)/(9) - (1)/(9) 7^2 x = (8)/(9) Divide by 7: ^2 x = (8)/(63) Step 3: Solve for x. Take the square root of both sides: x = ±sqrt((8)/(63)) Calculate the decimal value: x ≈ ±sqrt(0.126984) x ≈ ± 0.356348 Step 4: Find the reference angle. Let be the acute angle such that = 0.356348. = (0.356348) ≈ 20.87^ Step 5: Find all solutions for x in the range 0^ x < 360^. Case 1: x = 0.356348 (positive) Since x is positive, x lies in Quadrant I or Quadrant II. • In Quadrant I: x_1 = ≈ 20.87^ • In Quadrant II: x_2 = 180^ - ≈ 180^ - 20.87^ = 159.13^ Case 2: x = -0.356348 (negative) Since x is negative, x lies in Quadrant III or Quadrant IV. • In Quadrant III: x_3 = 180^ + ≈ 180^ + 20.87^ = 200.87^ • In Quadrant IV: x_4 = 360^ - ≈ 360^ - 20.87^ = 339.13^ Step 6: Round the solutions to the nearest degree. • x_1 ≈ 21^ • x_2 ≈ 159^ • x_3 ≈ 201^ • x_4 ≈ 339^ The solutions are 21^, 159^, 201^, 339^. Drop the next question.