\frac{\csc x}{5} - 2 = 0

Steps to Solve

1. Isolate the cosecant function

Start by moving the constant term to the other side of the equation.

$$ \frac{\csc x}{5} = 2 $$

2. Multiply by 5

To get rid of the fraction, multiply both sides of the equation by 5.

$$ \csc x = 10 $$

3. Convert to sine function

Recall that the cosecant function is the reciprocal of the sine function. Thus, we can rewrite the equation as:

$$ \sin x = \frac{1}{10} $$

4. Find the general solution

The sine function has a range of values between -1 and 1. Since $\sin x = \frac{1}{10}$, the general solutions can be expressed as:

$$ x = \arcsin\left(\frac{1}{10}\right) + 2k\pi \quad \text{and} \quad x = \pi - \arcsin\left(\frac{1}{10}\right) + 2k\pi \quad \text{for} ; k \in \mathbb{Z} $$

5. Calculate the specific angle

Using a calculator, we find:

$$ \arcsin\left(\frac{1}{10}\right) \approx 0.1001674 \text{ radians} $$

So,

$$ x \approx 0.1001674 + 2k\pi \quad \text{and} \quad x \approx \pi - 0.1001674 + 2k\pi $$

6. Simplify the second solution

Calculate the second solution:

$$ \pi - 0.1001674 \approx 3.0414252 $$

Thus the two solutions become:

$$ x \approx 0.1001674 + 2k\pi \quad \text{and} \quad x \approx 3.0414252 + 2k\pi $$