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Part 1
Find all values of theta if theta is in the interval ​[0degrees​,360degrees​) and has the given function value. 
tangent thetaalmost equals0.6117492
Question content area bottom
Part 1
The​ value(s) of theta ​is/are 
  
enter your response here.
​(Type an integer or decimal rounded to five decimal places as needed. Use a comma to separate answers as needed. Do not include the degree symbol in your​ answer.)

To find all values of \(\theta\) in the interval \([0^\circ, 360^\circ)\) such that \(\tan(\theta) \approx 0.6117492\), we need to solve for \(\theta\) where:

\[
\tan(\theta) = 0.6117492
\]

### Step 1: Find the Principal Value

To find the principal value, we use the inverse tangent function:

\[
\theta_1 = \arctan(0.6117492)
\]

Let's calculate this value first.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the calculation!

### Step-by-Step Solution

1. **Find the Principal Value**:  
   Use the inverse tangent function to find the first solution:
   \[
   \theta_1 = \arctan(0.6117492)
   \]

   You can calculate this using a calculator or software that provides the arctangent function. Make sure it is set to degrees mode.

2. **Find the Secondary Value**:  
   Since the tangent function has a period of \(180^\circ\), the general solutions are given by:
   \[
   \theta = \theta_1 + 180^\circ \times k
   \]
   where \(k\) is an integer.

   In the range \([0^\circ, 360^\circ)\), you will have two solutions:
   - \(\theta_1\)
   - \(\theta_2 = \theta_1 + 180^\circ\)

### Your Turn:
Calculate \(\arctan(0.6117492)\), round it to five decimal places, and then add \(180^\circ\) to find the second solution.

Would you like more details or help with the calculations? Here are some related questions to explore:

1. What is the tangent function's period and how does it affect the solution?
2. How can you use inverse trigonometric functions on a calculator?
3. Why are there two solutions for \(\theta\) in this interval?
4. What are the values of \(\tan(\theta)\) for standard angles?
5. How does changing the range of \(\theta\) affect the solutions?

**Tip:** When using trigonometric functions, always check if your calculator is set to degrees or radians mode, depending on the question requirements!

Find all values of theta if theta is in the interval [0 degrees, 360 degrees) and has the given function value. Tangent theta almost equals 0.6117492.

Learn how to find all values of theta in the interval [0°, 360°) where tan(θ) ≈ 0.6117492. This problem covers key trigonometric concepts, including inverse tangent functions and the periodicity of the tangent function. Suitable for students in Grades 10-12.

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