Do not round any intermediate computations. Give your answer(s) in radians, and round your answer(s) to the nearest hundredth.

To find the solutions of the equation \(\sin \theta = \frac{4}{5}\) in the interval \([0, 2\pi)\), we will proceed with exact values and round the final results to the nearest hundredth.

### Step 1: Identify the principal angle
The principal angle \(\theta_1\) is found by taking the inverse sine (arcsine) of \(\frac{4}{5}\):

\[
\theta_1 = \arcsin\left(\frac{4}{5}\right)
\]

Using a calculator:

\[
\theta_1 \approx \arcsin\left(\frac{4}{5}\right) \approx 0.9272952180016122 \text{ radians}
\]

### Step 2: Find the second solution
The second solution occurs in the second quadrant, where the sine function is also positive:

\[
\theta_2 = \pi - \theta_1
\]

Substituting the exact value of \(\pi\) and \(\theta_1\):

\[
\theta_2 = 3.141592653589793 - 0.9272952180016122 \approx 2.214297435588181 \text{ radians}
\]

### Step 3: Round the answers to the nearest hundredth
Now, we round both solutions to the nearest hundredth:

\[
\theta_1 \approx 0.93 \text{ radians}, \quad \theta_2 \approx 2.21 \text{ radians}
\]

### Final Answer
The solutions to the equation \(\sin \theta = \frac{4}{5}\) in the interval \([0, 2\pi)\) are:

\[
\theta \approx 0.93 \text{ radians}, \quad \theta \approx 2.21 \text{ radians}
\]

Would you like more details or have any questions?

### Related Questions:
1. How would you compute these angles using degrees instead of radians?
2. What are the sine values of the angles in other quadrants?
3. How does the accuracy of the final answer change when using fewer decimal places?
4. What if the equation were \(\sin \theta = \frac{1}{2}\)? How would the solutions differ?
5. How do we verify that our rounded solutions are accurate within the interval?

### Tip:
Always keep as many decimal places as possible in intermediate steps to ensure the final result is as accurate as needed when rounding.

Learn how to solve the trigonometric equation sin(theta) = 4/5 in radians. This detailed step-by-step guide covers finding principal angles, second solutions, and rounding answers to the nearest hundredth. Suitable for advanced high school students and anyone interested in trigonometry.

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