We are given the equation:

$2\sin^2x - \cos x = 1$

We need to solve this equation for $x$ in the interval $0^\circ \leq x \leq 360^\circ$.

Step 1: Substituting for $\sin^2 x$

We can use the Pythagorean identity: $\sin^2 x = 1 - \cos^2 x$ Substitute $\sin^2 x$ into the equation:

$2(1 - \cos^2 x) - \cos x = 1$

Step 2: Simplifying the equation

Expand and simplify:

$2(1) - 2\cos^2 x - \cos x = 1$

$2 - 2\cos^2 x - \cos x = 1$

Now, subtract 1 from both sides:

$1 - 2\cos^2 x - \cos x = 0$

Step 3: Rearranging the equation

Rearrange it as a quadratic equation in terms of $\cos x$:

$2\cos^2 x + \cos x - 1 = 0$

Step 4: Solving the quadratic equation

We can solve this quadratic equation using the quadratic formula. The quadratic formula is given by:

$\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $2\cos^2 x + \cos x - 1 = 0$, the coefficients are $a = 2$, $b = 1$, and $c = -1$. Substituting into the quadratic formula:

$\cos x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)}$ $\cos x = \frac{-1 \pm \sqrt{1 + 8}}{4}$ $\cos x = \frac{-1 \pm \sqrt{9}}{4}$ $\cos x = \frac{-1 \pm 3}{4}$

This gives us two solutions for $\cos x$:

1. $\cos x = \frac{-1 + 3}{4} = \frac{2}{4} = \frac{1}{2}$
2. $\cos x = \frac{-1 - 3}{4} = \frac{-4}{4} = -1$

Step 5: Finding the angles

Case 1: $\cos x = \frac{1}{2}$

The angles where $\cos x = \frac{1}{2}$ in the interval $0^\circ \leq x \leq 360^\circ$ are: $x = 60^\circ, 300^\circ$

Case 2: $\cos x = -1$

The angle where $\cos x = -1$ in the interval $0^\circ \leq x \leq 360^\circ$ is: $x = 180^\circ$

Step 6: Final solution

The solutions for $x$ are: $x = 60^\circ, 180^\circ, 300^\circ$

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. Solve $2\cos^2 x - \sin x = 0$ in the interval $0^\circ \leq x \leq 360^\circ$.
2. Solve $\sin 2x = \cos x$ in the same interval.
3. Find the general solution for $\sin^2 x - 2\sin x = 0$.
4. Solve $3\cos^2 x - 4\cos x + 1 = 0$ in $0^\circ \leq x \leq 360^\circ$.
5. Solve $\cos^2 x + \sin x = 1$ in the given interval.

Tip: When solving trigonometric equations, remember to consider all possible angles within the given interval that satisfy the equation!