To solve the equation $\tan(\theta) = 1.8$ on the interval $0 \leq \theta < 2\pi$, we need to identify the values of $\theta$ where the tangent of $\theta$ equals 1.8.

Step 1: Set up the equation

We are given:

$\tan(\theta) = 1.8$

This is a trigonometric equation where we need to find the values of $\theta$ that satisfy this condition within one full revolution (from $0$ to $2\pi$).

Step 2: Use the inverse tangent (arctan) function

To find the principal value of $\theta$, we can use the inverse tangent:

$\theta_1 = \tan^{-1}(1.8)$

Let’s compute this:

$\theta_1 \approx 1.071 \text{ radians}$

Step 3: General solution for the tangent function

The tangent function has a period of $\pi$, meaning that for any solution $\theta_1$, the general solutions are:

$\theta = \theta_1 + n\pi \quad \text{where} \quad n \in \mathbb{Z}$

Since we are asked to find solutions in the interval $0 \leq \theta < 2\pi$, we consider $n = 0$ and $n = 1$.

1. For $n = 0$, we get the first solution:

$\theta_1 = 1.071 \text{ radians}$

2. For $n = 1$, we get the second solution:

$\theta_2 = 1.071 + \pi \approx 1.071 + 3.1416 = 4.2126 \text{ radians}$

Step 4: Conclusion

Thus, the two solutions for $\tan(\theta) = 1.8$ in the interval $0 \leq \theta < 2\pi$ are:

$\theta_1 \approx 1.1 \text{ radians}$ $\theta_2 \approx 4.2 \text{ radians}$

These are the solutions to the equation in radians, rounded to the nearest tenth.

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Would you like more details on how the tangent function works, or do you have any other questions?

Here are some related questions you might find interesting:

1. What happens to the solutions if $\tan(\theta) = -1.8$?
2. How can you solve trigonometric equations using a unit circle?
3. What is the period of the sine and cosine functions?
4. How does the inverse tangent function relate to the unit circle?
5. How do you graph the tangent function and its periodicity?

Tip: Remember that the tangent function has vertical asymptotes at $\theta = \frac{\pi}{2} + n\pi$, where $n$ is an integer, so solutions that fall near these values must be carefully considered.