We are given the equation:

$3 \tan(\theta) - 8 = 0$

Step 1: Solve for $\tan(\theta)$

First, isolate $\tan(\theta)$:

$3 \tan(\theta) = 8$

$\tan(\theta) = \frac{8}{3}$

Step 2: Find the reference angle

Now, take the arctangent (inverse tangent) of both sides to find the reference angle:

$\theta_{\text{ref}} = \tan^{-1}\left(\frac{8}{3}\right)$

Using a calculator:

$\theta_{\text{ref}} \approx 69.4^\circ$

Step 3: Find all possible solutions

Since the tangent function is positive in both the first and third quadrants, we need to find the angles in both of these quadrants.

• First quadrant: $\theta_1 = 69.4^\circ$
• Third quadrant: $\theta_2 = 180^\circ + 69.4^\circ = 249.4^\circ$

Step 4: Final answers

The two solutions for $\theta$ in the range $0^\circ \leq \theta < 360^\circ$ are:

$\theta_1 \approx 69.4^\circ \quad \text{and} \quad \theta_2 \approx 249.4^\circ$

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Would you like more details or have any questions?
Here are 5 related questions to explore:

1. How do we handle tangent values outside typical quadrants?
2. What is the general solution for tangent equations?
3. How do we determine angles in different quadrants for sine and cosine?
4. How do we solve trigonometric equations involving cosine and sine?
5. What is the difference between inverse trigonometric functions?

Tip: When solving trigonometric equations, always consider the periodicity of the function to find all possible solutions.