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.
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multipleQuestions newQuestion solve for 1+tan^2 x=2tanx for 0<=x<=360 newAnswer The solutions for x are 45° and 225°. Steps: 1. Rearrange the equation: Start with the given equation: 1 + tan^2(x) = 2tan(x) Move all terms to one side to form a quadratic equation: 1 + tan^2(x) - 2tan(x) = 0 Rearrange in standard quadratic form ay^2 + by + c = 0, where y = tan(x): tan^2(x) - 2tan(x) + 1 = 0 2. Factor the quadratic equation: This is a perfect square trinomial, which can be factored as (a - b)^2 = a^2 - 2ab + b^2. Here, a = tan(x) and b = 1: (tan(x) - 1)^2 = 0 3. Solve for tan(x): Take the square root of both sides: tan(x) - 1 = 0 Add 1 to both sides: tan(x) = 1 4. Find the values of x in the given range (0° <= x <= 360°): The tangent function is positive in the first and third quadrants. First Quadrant: The angle whose tangent is 1 is 45°. x = 45° Third Quadrant: The angle in the third quadrant is 180° plus the reference angle. x = 180° + 45° = 225° Therefore, the solutions for x in the range 0° <= x <= 360° are 45° and 225°.