We are tasked with solving the equation:

$\tan(x) = 1.282$

for $x$ in the range $0^\circ \leq x < 360^\circ$. We need to identify all solutions and express them to three significant digits.

Step 1: Using the inverse tangent function

To find the principal solution, we take the inverse tangent of both sides:

$x = \tan^{-1}(1.282)$

This gives us the principal value for $x$. Using a calculator:

$x = \tan^{-1}(1.282) \approx 52.43^\circ$

Step 2: Identifying other solutions

The tangent function is periodic, with a period of $180^\circ$. This means that if $x = 52.43^\circ$ is a solution, the general solutions are:

$x = 52.43^\circ + 180^\circ \cdot n$

where $n$ is any integer. We now need to find all solutions between $0^\circ$ and $360^\circ$.

Step 3: Finding all solutions in the specified range

• For $n = 0$, we have $x = 52.43^\circ$.
• For $n = 1$, we have $x = 52.43^\circ + 180^\circ = 232.43^\circ$.

These are the only two solutions within the range $0^\circ \leq x < 360^\circ$.

Final Answer

Thus, the solutions to $\tan(x) = 1.282$ in the range $0^\circ \leq x < 360^\circ$ are:

$x = 52.4^\circ; 232.4^\circ$

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Would you like further details or have any questions? Here are some related questions you can explore:

1. How does the tangent function behave and why is it periodic with a period of 180°?
2. How do we solve trigonometric equations with other functions, such as sine or cosine?
3. What is the general form for solving trigonometric equations?
4. How can the unit circle help in understanding solutions to trigonometric equations?
5. What are some methods for verifying the solutions to trigonometric equations?

Tip: Always check that your solutions fall within the given range by adjusting for periodicity!