 10sin⁡xcos⁡x−5sin⁡x+4cos⁡x=210sinxcosx−5sinx+4cosx=2

To solve the equation:

\[
10 \sin x \cos x - 5 \sin x + 4 \cos x = 2
\]

we can start by using a trigonometric identity to simplify the expression. Recall the double angle identity:

\[
\sin(2x) = 2 \sin x \cos x
\]

Therefore, we can rewrite \(10 \sin x \cos x\) as \(5 \cdot 2 \sin x \cos x\), which simplifies to:

\[
5 \sin(2x)
\]

Substituting this into the original equation gives:

\[
5 \sin(2x) - 5 \sin x + 4 \cos x = 2
\]

Now, let's rearrange and factor out common terms:

\[
5 \sin(2x) - 5 \sin x + 4 \cos x - 2 = 0
\]

Divide the whole equation by 5 to simplify it:

\[
\sin(2x) - \sin x + \frac{4}{5} \cos x - \frac{2}{5} = 0
\]

Now, we need to find solutions for this equation. Let's solve it step-by-step:

### Step 1: Isolate \(\sin(2x)\)

\[
\sin(2x) = \sin x - \frac{4}{5} \cos x + \frac{2}{5}
\]

### Step 2: Introduce auxiliary variables or methods

This equation is quite complex due to the combination of trigonometric functions. We can proceed by considering specific values or using numerical methods for specific \(x\). Alternatively, since there is no straightforward way to analytically solve this equation (due to the mixture of angles \(x\) and \(2x\)), we may resort to a numerical or graphing approach.

### Step 3: Numerical/Graphical Solution

Using a graphing calculator or software like Desmos or a numerical method (Newton-Raphson, etc.), we could find approximate solutions for \(x\). Let's analyze particular values to check the equation:

1. Substitute \(x = 0\):

\[
10 \sin(0) \cos(0) - 5 \sin(0) + 4 \cos(0) = 4 \neq 2 \quad (\text{not a solution})
\]

2. Substitute \(x = \frac{\pi}{2}\):

\[
10 \sin\left(\frac{\pi}{2}\right) \cos\left(\frac{\pi}{2}\right) - 5 \sin\left(\frac{\pi}{2}\right) + 4 \cos\left(\frac{\pi}{2}\right) = -5 \neq 2 \quad (\text{not a solution})
\]

Further analysis would involve examining values in the domain of sine and cosine functions to see where the left-hand side equals the right-hand side.

### Step 4: General Solutions

Since we know that the equation is not easily factored, finding solutions by estimating values for \(x\) that satisfy the equation using numerical or graphing tools is appropriate. The solutions can be refined using trigonometric properties or by verifying roots within a certain interval numerically.

Would you like further assistance in solving this using a specific numerical method or graphing approach, or do you have any particular values of \(x\) you would like to test?

---

### Related Questions

1. How can we solve trigonometric equations involving mixed angles like \(x\) and \(2x\)?
2. What is the significance of the double-angle identity in simplifying trigonometric equations?
3. How does the Newton-Raphson method work for solving nonlinear equations?
4. What are some common strategies for solving non-factorable trigonometric equations?
5. How can graphing help visualize and solve complex trigonometric equations?

### Tip

For complex trigonometric equations, using trigonometric identities can simplify the problem. However, numerical methods or graphing tools are invaluable when identities do not provide a straightforward solution.

Learn how to solve the trigonometric equation 10sin⁡xcos⁡x−5sin⁡x+4cos⁡x=210sinxcosx−5sinx+4cosx=2 with detailed step-by-step solutions. This problem involves trigonometric identities and equations, specifically the double angle identity. Suitable for advanced high school students and those studying trigonometry.

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