Solve for x 10sin(x)^2=10+5cos(x)

Problem

10*sin2(x)=10+5*cos(x)

Solution

1. Apply the Pythagorean identity to rewrite the equation in terms of a single trigonometric function. Substitute sin2(x)=1−cos2(x) into the equation.

10*(1−cos2(x))=10+5*cos(x)

2. Distribute the constant on the left side.

10−10*cos2(x)=10+5*cos(x)

3. Rearrange the equation into a standard quadratic form by moving all terms to one side and simplifying. Subtract 10 from both sides.

−10*cos2(x)=5*cos(x)

4. Set the equation to zero by adding 10*cos2(x) to both sides.

0=10*cos2(x)+5*cos(x)

5. Factor out the greatest common factor, which is 5*cos(x)

5*cos(x)*(2*cos(x)+1)=0

6. Apply the Zero Product Property to find the possible values for cos(x)

5*cos(x)=0⇒cos(x)=0

2*cos(x)+1=0⇒cos(x)=−1/2

7. Solve for x using the unit circle. For cos(x)=0 the solutions are x=π/2+n*π For cos(x)=−1/2 the solutions are x=(2*π)/3+2*n*π and x=(4*π)/3+2*n*π

Final Answer

x=π/2+n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π

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