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We are given that:

\[
\cos(\theta) \approx 0.4052056
\]

We need to find all possible values of \(\theta\) in the interval \([0^\circ, 360^\circ)\).

To solve for \(\theta\):

1. First, take the inverse cosine (arccos) of \(0.4052056\) to find one solution.
2. Since cosine is positive in both the first and fourth quadrants, the second solution will be in the fourth quadrant.

Let's calculate the two solutions. I will compute these now.The two solutions for \(\theta\) in the interval \([0^\circ, 360^\circ)\) are approximately:

\[
\theta_1 \approx 66^\circ
\]
\[
\theta_2 \approx 294^\circ
\]

Thus, the values of \(\theta\) are \(66^\circ\) and \(294^\circ\).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would you find the values if \(\cos(\theta)\) was negative?
2. Can you explain why there are two solutions for \(\theta\) in the range \([0^\circ, 360^\circ)\)?
3. How would you solve this if you were working in radians instead of degrees?
4. What are the general solutions for \(\theta\) for \(\cos(\theta) = 0.4052056\)?
5. What happens to the solutions if the interval is extended to multiple cycles of \(360^\circ\)?

**Tip:** Always remember that cosine is positive in both the first and fourth quadrants when solving trigonometric equations.

Find all values of θ if θ is in the interval [0°, 360°) and has the given function value: cos(θ) ≈ 0.4052056.

Discover how to find all values of θ in the interval [0°, 360°) for the given function value cos(θ) ≈ 0.4052056. This trigonometry problem explores the cosine function and its properties in the unit circle. Perfect for high school students in Grades 10-12.

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