Respuesta :
Given:
[tex]cos(2x)=1-sin(x)[/tex]
Expanding using double angle formula,
[tex]1 - 2 \sin^{2} (x) = 1 - \sin(x) [/tex]
Rearranging into a single equation,
[tex]1 - 2 \sin^{2}(x) - 1 + \sin(x) = 0[/tex]
Combining like terms,
[tex] - 2 \sin^{2} (x) + \sin(x) = 0[/tex]
Now multiplying each term by -1,
[tex]2 \sin^{2} (x) - \sin(x) = 0[/tex]
Factor greatest common factors out,
[tex] \sin(x)(2 \sin(x) - 1) = 0[/tex]
Applying Zero property rule,
[tex] \sin(x) = 0 \: or \: 2 \sin(x) - 1 = 0[/tex]
{ For sin(x) = 0
• Solve the trignometric equation to find a particular solution,
[tex]x = 0 \: or \: x = π[/tex]
[tex] = > x = 2nπ \: or \: x = π + 2nπ[/tex]
• Find the union of the solution sets,
[tex]x = nπ[/tex]}
{ For 2 sin(x) - 1 = 0 ,
• Rearrange unknown terms to the left side,
[tex]2 \sin(x) = 1[/tex]
• Divide both sides of the equation by the coefficient of variable,
[tex] \sin(x) = \frac{1}{2} [/tex]
• Solve the trignometric equation to find a particular solution,
[tex]x = \frac{π}{6} \: or \: x = \frac{5π}{6} [/tex]
[tex] = > x = \frac{π}{6} + 2nπ \: or \: x = \frac{5π}{6} + 2nπ,n∈Z[/tex]
}
☆ Now finding the union of both sets
[tex]x = nπ \: or \: x = \frac{π}{6} + 2nπ \: or \: x = \frac{5π}{6} + 2nπ,n∈Z[/tex]
Hence, the answer is [tex]x = nπ \: or \: x = \frac{π}{6} + 2nπ \: or \: x = \frac{5π}{6} + 2nπ,n∈Z[/tex]
