To solve the equation:

\[
\frac{5}{\tan(x)} + \sin(x) = 0
\]

First, recall that:

\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\]

Substituting this into the equation:

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

This simplifies to:

\[
5 \cdot \frac{\cos(x)}{\sin(x)} + \sin(x) = 0
\]

Multiply both terms by \(\sin(x)\) to eliminate the denominator:

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

Now isolate \( \cos(x) \):

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

Since \(\sin^2(x) + \cos^2(x) = 1\), we can substitute \( \sin^2(x) \) with \( 1 - \cos^2(x) \):

\[
5 \cos(x) = -(1 - \cos^2(x))
\]

Expanding the equation:

\[
5 \cos(x) = -1 + \cos^2(x)
\]

Rearrange everything to one side:

\[
\cos^2(x) - 5 \cos(x) - 1 = 0
\]

This is now a quadratic equation in terms of \( \cos(x) \). Let \( y = \cos(x) \), so the equation becomes:

\[
y^2 - 5y - 1 = 0
\]

We can solve this quadratic using the quadratic formula:

\[
y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-1)}}{2(1)}
\]

Simplifying:

\[
y = \frac{5 \pm \sqrt{25 + 4}}{2}
\]
\[
y = \frac{5 \pm \sqrt{29}}{2}
\]

Thus, the solutions for \( y \) (or \( \cos(x) \)) are:

\[
\cos(x) = \frac{5 + \sqrt{29}}{2} \quad \text{or} \quad \cos(x) = \frac{5 - \sqrt{29}}{2}
\]

However, since \(\cos(x)\) must be between \(-1\) and \(1\), we discard the solution \(\cos(x) = \frac{5 + \sqrt{29}}{2}\), because it is greater than 1.

So, we are left with:

\[
\cos(x) = \frac{5 - \sqrt{29}}{2}
\]

Now, to find the specific solutions for \(x\), use the inverse cosine function:

\[
x = \cos^{-1}\left(\frac{5 - \sqrt{29}}{2}\right)
\]

You can find the exact value of this expression using a calculator.

Would you like to see the steps in more detail or have any questions?

Here are 5 related questions to consider:
1. What is the domain and range of the cosine function?
2. How can we verify if a solution for \(x\) falls within the standard trigonometric interval?
3. What other trigonometric identities can simplify equations involving sine and cosine?
4. How would this equation change if the \(\sin(x)\) term were \(\cos(x)\) instead?
5. What are the potential uses of solving trigonometric equations in real-world applications?

**Tip**: Always check whether the solutions fall within the allowed range for trigonometric functions, especially for cosine and sine, which are restricted between -1 and 1.

Explore how to solve the trigonometric equation ((5/tan(x)) + sin(x) = 0) with step-by-step guidance. The solution involves applying trigonometric identities, simplifying into a quadratic equation, and solving using the quadratic formula. Perfect for high school students (Grades 10-12).

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