3 sin ² х +7 cos x -3=0

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To solve the equation 3sin^2(x) + 7cos(x) - 3 = 0, we can use some trigonometric identities to simplify the expression.

Start by using the Pythagorean Identity: sin^2(x) + cos^2(x) = 1.

Rewrite the equation in terms of sin^2(x) only:
3sin^2(x) = 3 - 3cos^2(x).

Substitute this into the original equation:
3(3 - 3cos^2(x)) + 7cos(x) - 3 = 0
9 - 9cos^2(x) + 7cos(x) - 3 = 0
-9cos^2(x) + 7cos(x) + 6 = 0

This is now a quadratic equation in terms of cos(x). Let y = cos(x) for convenience:

-9y^2 + 7y + 6 = 0

Now, you can solve this quadratic equation using the quadratic formula or factoring. Once you find the solutions for y (cos(x)), you can then solve for x by taking the arccosine of those values. Remember to verify your solutions within the given range of the trigonometric function.