https://file.aiquickdraw.com/mathbot/file/1727230466309w9ewod0n.jpg

To find all the values of \(\theta\) such that \(\sin(\theta) \approx 0.4050197\) in the interval \([0^\circ, 360^\circ)\), we can follow these steps:

### Step 1: Find the reference angle
The reference angle for a sine value is calculated using the inverse sine (arcsin) function:

\[
\theta_{\text{ref}} = \arcsin(0.4050197)
\]

Let's calculate the reference angle.

### Step 2: Determine other solutions
Since sine is positive in the first and second quadrants, we will have two solutions:
1. The first solution is the reference angle.
2. The second solution is given by \(180^\circ - \theta_{\text{ref}}\).

Let's compute these values now.The two solutions for \(\theta\) in the interval \([0^\circ, 360^\circ)\) where \(\sin(\theta) \approx 0.4050197\) are:

- \(\theta_1 \approx 23.89236^\circ\)
- \(\theta_2 \approx 156.10764^\circ\)

Would you like further details or explanations? 

Here are five related questions for further exploration:

1. How is the inverse sine function used to find angles?
2. What is the general solution for sine equations in trigonometry?
3. How does the sine function behave in different quadrants?
4. Can you calculate angles for other trigonometric values like cosine or tangent?
5. What are the uses of reference angles in solving trigonometric equations?

**Tip:** When solving trigonometric equations, remember that the sine function is positive in the first and second quadrants.

Find all values of θ if θ is in the interval [0°, 360°) and has the given function value sin(θ) ≈ 0.4050197.

Find the values of θ for which sin(θ) ≈ 0.4050197 in the interval [0°, 360°). This trigonometry problem demonstrates how to calculate angles using the sine function and reference angles. Suitable for high school students studying Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation