To solve the equation $12 \cos^2 x - \sin x = 11$, let's go step by step.

Step 1: Use the Pythagorean identity

We know that $\cos^2 x + \sin^2 x = 1$. So, we can express $\cos^2 x$ in terms of $\sin x$.

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

Step 2: Substitute into the equation

Now, substitute $\cos^2 x = 1 - \sin^2 x$ into the original equation:

$12 (1 - \sin^2 x) - \sin x = 11$

Step 3: Simplify the equation

Distribute the 12 on the left side:

$12 - 12 \sin^2 x - \sin x = 11$

Now, subtract 11 from both sides:

$12 - 12 \sin^2 x - \sin x - 11 = 0$

This simplifies to:

$1 - 12 \sin^2 x - \sin x = 0$

Step 4: Rearrange the equation

Now, we have a quadratic equation in terms of $\sin x$:

$-12 \sin^2 x - \sin x + 1 = 0$

Multiply through by -1 to simplify:

$12 \sin^2 x + \sin x - 1 = 0$

Step 5: Solve the quadratic equation

Now we have a quadratic equation in $\sin x$:

$12 \sin^2 x + \sin x - 1 = 0$

We can solve this using the quadratic formula:

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

where $a = 12$, $b = 1$, and $c = -1$. Substituting these values into the formula:

$\sin x = \frac{-1 \pm \sqrt{1^2 - 4(12)(-1)}}{2(12)}$

$\sin x = \frac{-1 \pm \sqrt{1 + 48}}{24}$

$\sin x = \frac{-1 \pm \sqrt{49}}{24}$

$\sin x = \frac{-1 \pm 7}{24}$

Step 6: Solve for the two possible values of $\sin x$

1. $\sin x = \frac{-1 + 7}{24} = \frac{6}{24} = \frac{1}{4}$
2. $\sin x = \frac{-1 - 7}{24} = \frac{-8}{24} = -\frac{1}{3}$

Step 7: Find the corresponding $x$ values

We now have two possible values for $\sin x$:

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

We can solve for $x$ in each case using inverse sine.

1. $x = \sin^{-1} \left(\frac{1}{4}\right)$
2. $x = \sin^{-1} \left(-\frac{1}{3}\right)$

Let me know if you'd like me to calculate these specific values or if you'd like more details on any of the steps!

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Here are 5 related questions to expand on this:

1. How do you solve quadratic equations with trigonometric functions?
2. How do you apply the Pythagorean identity to simplify trigonometric equations?
3. What are the general solutions for trigonometric equations like $\sin x = \frac{1}{4}$ or $\sin x = -\frac{1}{3}$?
4. How does the inverse sine function help in finding the angle corresponding to a sine value?
5. What are the possible solutions for trigonometric equations on different intervals, such as $[0, 2\pi]$?

Tip: When solving trigonometric equations, always remember to consider multiple possible solutions within the required interval.