Find all solutions of the equation in the interval [0,2pi ) -4cosx=-sin^2x+1 Write your answer in radians in terms of pi If there is more than one solution separate them with commas. x=[ ]

Question

Answer 1

To solve the equation

in the interval \([0, 2\pi)\), we can follow these steps:1. Rewrite the equation:

2. Use the Pythagorean identity

to express

in terms of

:

3. Substitute

into the equation:

4. Square both sides to eliminate the square root:

5. Multiply both sides by 16 to clear the fraction:

6. Expand the right-hand side:

7. Rearrange the equation to form a quadratic in

:

8. Let

, then the equation becomes:

9. Solve the quadratic equation using the quadratic formula

:

10. Since

must be in the interval

, we need to check which solutions are valid:

Only

is in the interval

.11. Find

from

:

12. Determine the corresponding

values in the interval \([0, 2\pi)\):

Thus, the solutions in radians in terms of

are:

Question