To solve the equation [tex]\(-\cos x = -\sin^2 x - 1\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Simplify the equation:
The given equation is:
[tex]\[
-\cos x = -\sin^2 x - 1
\][/tex]
Remove the negative signs by multiplying both sides by [tex]\(-1\)[/tex]:
[tex]\[
\cos x = \sin^2 x + 1
\][/tex]
2. Use the Pythagorean identity:
Recall the Pythagorean identity:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Substitute [tex]\(\sin^2 x\)[/tex] with [tex]\(1 - \cos^2 x\)[/tex]:
[tex]\[
\cos x = 1 - \cos^2 x + 1
\][/tex]
3. Simplify further:
Combine like terms:
[tex]\[
\cos x = 2 - \cos^2 x
\][/tex]
Rearrange the equation to form a quadratic equation:
[tex]\[
\cos^2 x + \cos x - 2 = 0
\][/tex]
4. Solve the quadratic equation:
Let [tex]\(u = \cos x\)[/tex]. The equation becomes:
[tex]\[
u^2 + u - 2 = 0
\][/tex]
Factorize the quadratic equation:
[tex]\[
(u + 2)(u - 1) = 0
\][/tex]
This gives us two solutions for [tex]\(u\)[/tex]:
[tex]\[
u + 2 = 0 \quad \Rightarrow \quad u = -2
\][/tex]
[tex]\[
u - 1 = 0 \quad \Rightarrow \quad u = 1
\][/tex]
Since [tex]\(\cos x\)[/tex] ranges from -1 to 1, [tex]\(u = -2\)[/tex] is not a feasible solution. Thus, we have:
[tex]\[
\cos x = 1
\][/tex]
5. Find the values of [tex]\(x\)[/tex]:
The cosine of [tex]\(x\)[/tex] is equal to 1 at:
[tex]\[
x = 0 \quad \text{and} \quad x = 2\pi
\][/tex]
6. Limit the solutions to the interval [tex]\([0, 2\pi)\)[/tex]:
In the interval [tex]\([0, 2\pi)\)[/tex], the only valid solution is:
[tex]\[
x = 0
\][/tex]
So, the solutions to the equation [tex]\(-\cos x = -\sin^2 x - 1\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
x = 0, \; 2\pi
\][/tex]
To present the answer in radians in terms of [tex]\(\pi\)[/tex]:
[tex]\[
x = 0, 2\pi
\][/tex]
