To solve for the angles $\theta$ that satisfy the equation:

$\tan(\theta) = \frac{5}{4}$

we will follow these steps:

Step 1: Use the inverse tangent (arctan) function

The inverse tangent will give us the principal value of $\theta$. So, we calculate:

$\theta_1 = \tan^{-1}\left(\frac{5}{4}\right)$

Using a calculator:

$\theta_1 \approx 51.34^\circ$

This is the first solution in the range $0^\circ \leq \theta < 360^\circ$.

Step 2: Find the second solution

The tangent function has a period of $180^\circ$, so another solution will occur in the second quadrant. This solution can be found by adding $180^\circ$ to the first solution:

$\theta_2 = 51.34^\circ + 180^\circ = 231.34^\circ$

Final Answer:

Thus, the two angles $\theta$ that satisfy the equation $\tan(\theta) = \frac{5}{4}$ in the range $0^\circ \leq \theta < 360^\circ$ are:

$\theta \approx 51.3^\circ \quad \text{and} \quad \theta \approx 231.3^\circ$

Summary:

• $\theta_1 \approx 51.3^\circ$
• $\theta_2 \approx 231.3^\circ$

Would you like any further clarification or detailed steps?

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Here are 5 related questions for further exploration:

1. What is the general form of the solution for $\tan(\theta) = k$, where $k$ is a constant?
2. How does the period of the tangent function affect the number of solutions in different intervals?
3. How can you use a calculator to find the inverse trigonometric functions for other angles?
4. What is the relationship between the reference angle and the solutions for trigonometric equations?
5. How do solutions for trigonometric functions differ in different quadrants?

Tip: When solving trigonometric equations, always check the quadrant in which the function is positive or negative to identify all possible solutions.