Solve for all values of theta*θ*, such that 0, degrees, is less than or equal to, theta, is less than, 360, degrees0∘≤*θ*<360∘, rounding all values *to the nearest tenth*.

3, tangent, theta, minus, 8, equals, 0

3tan*θ*−8=

0

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:

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

---

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.

Solve for all values of theta, such that 0° ≤ θ < 360°, rounding all values to the nearest tenth. 3tan(θ) - 8 = 0

Discover how to solve the equation 3tan(θ) - 8 = 0 for values of theta between 0° and 360°, using trigonometric concepts like the inverse tangent and understanding of angle quadrants. This step-by-step solution is designed for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation