Question - Solving Trigonometric Equation with Quadratic Formula
Solution:
The equation given is trigonometric, and it can be rewritten and solved by exploiting the trigonometric identities.The equation is:sec^2(x) - 2tan(x) - 2 = 0Now, considering the trigonometric identity sec^2(x) = tan^2(x) + 1, let's rewrite the equation using this identity:tan^2(x) + 1 - 2tan(x) - 2 = 0Simplify it:tan^2(x) - 2tan(x) - 1 = 0This is a quadratic equation in tan(x). To solve for tan(x), we use the quadratic formula: if ax^2 + bx + c = 0, then x = (-b ± √(b^2 - 4ac))/(2a).Here, a = 1, b = -2, and c = -1.Using the quadratic formula:tan(x) = (-(-2) ± √((-2)^2 - 4*1*(-1)))/(2*1)tan(x) = (2 ± √(4 + 4))/2tan(x) = (2 ± √8)/2tan(x) = (2 ± 2√2)/2tan(x) = 1 ± √2This gives us two possible values for tan(x):tan(x) = 1 + √2 or tan(x) = 1 - √2To find x, we look for the angle whose tangent is these values. The exact values of the angles depend on the interval for x that we are solving over, which has not been provided. In general, you would use a calculator or inverse tangent function to find these angles within a given interval. Keep in mind the periodic nature of the tangent function and the fact that it is positive in the first and third quadrants and negative in the second and fourth quadrants if you are working with standard intervals like [0, 2π) or [-π, π).
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