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Find all solutions of the equation. sec² x - 2 = 0

User Mina Samir
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1 Answer

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Final answer:

To solve sec² x - 2 = 0, we take the square root after isolating sec² x to find sec x = ±√2, then find the corresponding cosine values and their angles, keeping in mind the periodicity of the trigonometric functions.

Step-by-step explanation:

The question asks to find all solutions of the equation sec² x - 2 = 0. This equation can be solved by first isolating sec² x, giving us sec² x = 2. Taking the square root of both sides, we get sec x = ±√2. Since secant is the reciprocal of cosine, we have cos x = ±√(1/2). The solutions for x are the angles where the cosine takes the value of ±√(1/2). These angles can be obtained from the unit circle or using trigonometric identities. For cos x = √(1/2), the solutions are x = ±45° + 360°k and x = 315° + 360°k. For cos x = -√(1/2), the solutions are x = 135° + 360°k and x = 225° + 360°k, where k is any integer representing the periodic nature of the cosine function.

User Root
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