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The problem asks to find all solutions to the equation:

\[
\sin(\theta) = 0.5956
\]

in the interval \([0, 2\pi]\) and provide the solutions in radians, rounded to the nearest hundredth. Let's solve this step by step.

### Step 1: Solve for the reference angle
To find the principal solution, we apply the inverse sine function:

\[
\theta_1 = \sin^{-1}(0.5956)
\]

Calculating this gives the reference angle.

### Step 2: Find additional solutions
Since the sine function is positive in the first and second quadrants, we will have two solutions in the interval \([0, 2\pi]\):
- One in the first quadrant: \(\theta_1\).
- One in the second quadrant: \(\theta_2 = \pi - \theta_1\).

### Step 3: Final solutions in the interval \([0, 2\pi]\)
The two solutions are \(\theta_1\) and \(\theta_2\).

Let me calculate these values now.The solutions to the equation \(\sin(\theta) = 0.5956\) in the interval \([0, 2\pi]\) are:

\[
\theta = 0.64 \, \text{radians}, \, 2.50 \, \text{radians}
\]

These are rounded to the nearest hundredth, as required.

Would you like more details on this solution or have any further questions? Here are a few related questions:

1. What is the general solution for \(\sin(\theta) = 0.5956\) outside the interval \([0, 2\pi]\)?
2. How does the sine function behave in different quadrants?
3. Can this equation have more than two solutions within the interval \([0, 2\pi]\)?
4. What would the solutions be if \(\sin(\theta) = -0.5956\)?
5. How do you derive solutions for cosine or tangent equations?

**Tip:** The sine function is periodic, which means it repeats its values over regular intervals. For \(\sin(\theta) = 0.5956\), you can find an infinite number of solutions by adding multiples of \(2\pi\) to the found solutions.

Hallar todas las soluciones de la ecuación siguiente en el intervalo [0, 2π]: sin(θ) = 0.5956

Find all the solutions to the trigonometric equation sin(θ) = 0.5956 in radians within the interval [0, 2π]. This step-by-step guide covers key concepts in trigonometry, including the sine function, inverse sine, and properties of sine in different quadrants. Suitable for high school students in Grades 10-12.

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