trigonometry problem

Question

Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."

−4cos2(x)−7cos(x)=3

Answer 1

since this is a quadratic will start by setting to zero ( subtract 3 from both sides and multiple both sides by -1 gives you 4cos^2(x) +7cos(x) +3 = 0 now factor to

(4cos(x)+ 3) ( cos(x)+1) =0 so solutions are

4cos(x)=-3 cos(x)-3/4 take inverse cosine of both sides an you get x = 2.4188 radians and x=-2.4188

cos is negative in both quadrant 2 and 3

The other solution is cos x =-1 which would be at pi