Find all solutions of the equation in the interval [0, 2π) algebraically.

Question

Find all solutions of the equation in the interval [0, 2π) algebraically. Use the table feature of a graphing utility to check your answers numerically. (List the solutions in increasing order. If there is no solution, enter NO SOLUTION.)

8 sin² x = 8 − 4 cos x

x=_______lowest value

x=_______

x=_______

x=_______ greatest value

Answer 1

8 sin² x = 8 − 4 cos x

since sin² x = 1 - cos² x, then

8 (1 - cos² x) = 8 − 4 cos x

8 - 8 cos² x = 8 − 4 cos x

8 cos² x − 4 cos x = 0

4 cos x ( 2 cos x - 1) = 0

4 cos x = 0, 2 cos x - 1 = 0

cos x = 0, cos x = 1/2

x = π/2, 3π/2, π/3, 5π/3

x = π/3, π/2, 3π/2, 5π/3 (from lowest value to greatest value)