To solve the equation $\tan \theta = 1.782$ for $0^\circ \leq \theta < 360^\circ$, we proceed as follows:

Step 1: Use the arctangent function

$\theta_1 = \tan^{-1}(1.782)$

Using a calculator: $\theta_1 \approx 60.98^\circ$

Step 2: Identify all solutions within one period

The tangent function has a period of $180^\circ$, meaning the general solutions are given by: $\theta = \theta_1 + 180^\circ k, \quad k \in \mathbb{Z}$

For $0^\circ \leq \theta < 360^\circ$, we calculate:

• $\theta_1 = 60.98^\circ$
• Adding $180^\circ$: $\theta_2 = 60.98^\circ + 180^\circ = 240.98^\circ$

Step 3: Verify the solutions

Both $\theta_1 \approx 60.98^\circ$ and $\theta_2 \approx 240.98^\circ$ satisfy $\tan \theta = 1.782$.

Final Answer:

The angles are: $\boxed{\theta \approx 60.98^\circ \text{ and } 240.98^\circ}$

Would you like further clarification or details?

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Here are 5 related questions for further exploration:

1. What is the general solution for $\tan \theta = k$?
2. How does the periodicity of the tangent function affect the solution set?
3. How do you solve similar equations for $\sin \theta$ or $\cos \theta$?
4. How does rounding impact the accuracy of trigonometric solutions?
5. How would the solutions change if $\tan \theta = -1.782$?

Tip: Always verify solutions using the original equation to ensure accuracy, especially when dealing with approximations.