To solve the trigonometric equation \( 2\cos^2x + 5\sin x = 4 \), we can proceed with the following steps:

### Step 1: Express \( \cos^2x \) in terms of \( \sin x \)
Using the Pythagorean identity:
\[
\cos^2x = 1 - \sin^2x
\]
Substitute this into the equation:
\[
2(1 - \sin^2x) + 5\sin x = 4
\]
Simplify the equation:
\[
2 - 2\sin^2x + 5\sin x = 4
\]

### Step 2: Rearrange the equation
Move all terms to one side to set the equation to zero:
\[
-2\sin^2x + 5\sin x + 2 - 4 = 0
\]
\[
-2\sin^2x + 5\sin x - 2 = 0
\]

### Step 3: Solve the quadratic equation
Multiply the entire equation by \(-1\) to make the quadratic coefficient positive:
\[
2\sin^2x - 5\sin x + 2 = 0
\]
Let \( u = \sin x \). The equation becomes:
\[
2u^2 - 5u + 2 = 0
\]
This is a standard quadratic equation. Solve it using the quadratic formula:
\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 2 \), \( b = -5 \), and \( c = 2 \):
\[
u = \frac{5 \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}
\]
\[
u = \frac{5 \pm \sqrt{25 - 16}}{4}
\]
\[
u = \frac{5 \pm \sqrt{9}}{4}
\]
\[
u = \frac{5 \pm 3}{4}
\]

So, the solutions for \( u \) are:
\[
u = \frac{8}{4} = 2 \quad \text{(not valid, since } \sin x \text{ must be between } -1 \text{ and } 1\text{)}
\]
\[
u = \frac{2}{4} = \frac{1}{2}
\]

### Step 4: Solve for \( x \)
Now, we solve for \( x \) when \( \sin x = \frac{1}{2} \).

The general solution for \( \sin x = \frac{1}{2} \) is:
\[
x = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{6} + 2n\pi
\]
where \( n \) is any integer.

### Final Answer:
The solutions are:
\[
x = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{6} + 2n\pi \quad \text{for any integer } n.
\]

Would you like more details, or do you have any questions?

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**Here are 5 related questions:**

1. How do you solve a quadratic equation using the quadratic formula?
2. What are the general solutions for basic trigonometric equations like \( \sin x = a \)?
3. How can you use trigonometric identities to simplify complex equations?
4. What are the domain and range of the sine function?
5. How do you solve trigonometric equations involving multiple angles?

**Tip:** When solving trigonometric equations, always consider the range of the trigonometric functions involved to ensure the solutions are valid.

Learn how to solve the trigonometric equation 2cos^2x + 5sinx = 4 with a detailed step-by-step solution. This problem involves trigonometric identities and the quadratic formula, suitable for advanced high school students.

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