Find all solutions of the equation in the interval [0, 2pi).    sin 0 = 4/5

To find the solutions of the equation \(\sin \theta = \frac{4}{5}\) in the interval \([0, 2\pi)\), we proceed as follows:

### Step 1: Identify the principal angle
The equation \(\sin \theta = \frac{4}{5}\) suggests that \(\theta\) is in either the first or the second quadrant because the sine function is positive in these quadrants.

To find the principal angle, we take the inverse sine (arcsine) of \(\frac{4}{5}\):

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

Using a calculator (in radians mode), this gives:

\[
\theta_1 \approx \arcsin\left(\frac{4}{5}\right) \approx 0.9273 \text{ radians}
\]

### Step 2: Find the second solution
The sine function is symmetric about \(\frac{\pi}{2}\) in the interval \([0, \pi]\). Therefore, the second solution in the interval \([0, 2\pi)\) is given by:

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

\[
\theta_2 \approx \pi - 0.9273 \approx 2.2143 \text{ radians}
\]

### Step 3: Write the solutions
The solutions to the equation \(\sin \theta = \frac{4}{5}\) in the interval \([0, 2\pi)\) are:

\[
\theta \approx 0.9273 \text{ radians}, \quad \theta \approx 2.2143 \text{ radians}
\]

These are the two angles within the given interval.

Would you like more details or have any questions?

### Related Questions:
1. How do we convert these angles from radians to degrees?
2. What is the general solution for \(\sin \theta = \frac{4}{5}\)?
3. How does the sine function behave in different quadrants?
4. What are the properties of the inverse sine function?
5. How would the solutions change if the equation were \(\sin \theta = -\frac{4}{5}\)?

### Tip:
Always remember that the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. This helps in identifying the correct angles when solving trigonometric equations.

Learn how to find solutions of the equation sin(theta) = 4/5 in the interval [0, 2pi). Step-by-step solution includes identifying principal angles, using inverse sine function, and applying symmetry of sine function. Suitable for high school students studying trigonometry.

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