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Find all solutions in the interval [0,2π) for the equation 2cos²(x)+3cos(x)=−1.

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Final answer:

We treat the equation as a quadratic by letting u = cos(x), solve for u, and then determine the corresponding angles x where cosine equals the solutions for u within the interval [0,2π).

Step-by-step explanation:

To solve the equation 2cos²(x)+3cos(x)=-1 in the interval [0,2π), we can treat it as a quadratic equation by setting u = cos(x). The equation then becomes 2u² + 3u + 1 = 0.

By factoring this quadratic equation or using the quadratic formula, we can find the values of u which are the cosine of the angle x.

Once we have the possible values for cos(x), we can find the corresponding angle x within the given interval, remembering the periodic nature of the cosine function and its range from -1 to 1.

For example, if we find that u = cos(x) = -1/2, then the possible angles could be 2π/3 and 4π/3 in the interval [0,2π), since cosine is negative in the second and third quadrants.

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