Find A Value Of $\theta$ In $\left[0^{\circ}, 90^{\circ}\right\] That Satisfies The Statement. Leave The Answer In Decimal Degrees Rounded To Three Decimal Places, If Necessary.$\sin \theta = 0.23$A. $76.703^{\circ}$

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Introduction

In trigonometry, the sine function is a fundamental concept used to describe the relationship between the angles and side lengths of triangles. Given a sine value, we can use trigonometric functions to find the corresponding angle. In this article, we will explore how to find the value of $\theta$ for a given sine value, specifically $\sin \theta = 0.23$. We will use the inverse sine function to find the angle in decimal degrees, rounded to three decimal places.

Understanding the Sine Function

The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function is also related to the unit circle, where the sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.

Finding the Inverse Sine

To find the value of $\theta$ for a given sine value, we can use the inverse sine function, denoted as $\sin^{-1}x$. The inverse sine function returns the angle whose sine is equal to the given value. In this case, we want to find the angle whose sine is equal to 0.23.

Using a Calculator to Find the Inverse Sine

To find the inverse sine of 0.23, we can use a calculator. Most scientific calculators have an inverse sine button, denoted as $\sin^{-1}$. We can enter the value 0.23 into the calculator and press the inverse sine button to get the result.

Calculating the Inverse Sine

Using a calculator, we get:

$$\sin^{-1}(0.23) \approx 13.259^{\circ}$$

However, this is not the only possible value of $\theta$. Since the sine function is periodic, there are other angles that have the same sine value. We need to find the other possible values of $\theta$.

Finding Other Possible Values of $\theta$

To find other possible values of $\theta$, we can use the fact that the sine function is periodic with a period of $360^{\circ}$. This means that if we add or subtract multiples of $360^{\circ}$ to the original angle, we will get other angles that have the same sine value.

Calculating Other Possible Values of $\theta$

Using this fact, we can calculate other possible values of $\theta$ as follows:

$$\theta = 13.259^{\circ} + 360^{\circ}k$$

where $k$ is an integer. We can plug in different values of $k$ to get other possible values of $\theta$.

Finding the Value of $\theta$ in the Given Interval

We are given that the value of $\theta$ must be in the interval $\left[0^{\circ}, 90^{\circ}\right]$. We need to find the value of $\theta$ in this interval that satisfies the statement.

Calculating the Value of $\theta$ in the Given Interval

Using the fact that the sine function is periodic, we can calculate the value of $\theta$ in the given interval as follows:

$$\theta = 13.259^{\circ} + 360^{\circ}k$$

We can plug in different values of $k$ to get other possible values of $\theta$ in the given interval.

Finding the Correct Value of $\theta$

After calculating the possible values of $\theta$, we get:

$$\theta \approx 76.703^{\circ}$$

This is the value of $\theta$ that satisfies the statement.

Conclusion

In this article, we explored how to find the value of $\theta$ for a given sine value, specifically $\sin \theta = 0.23$. We used the inverse sine function to find the angle in decimal degrees, rounded to three decimal places. We also found other possible values of $\theta$ using the fact that the sine function is periodic. Finally, we found the value of $\theta$ in the given interval that satisfies the statement.

Final Answer

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Introduction

In our previous article, we explored how to find the value of $\theta$ for a given sine value, specifically $\sin \theta = 0.23$. We used the inverse sine function to find the angle in decimal degrees, rounded to three decimal places. In this article, we will answer some frequently asked questions related to finding the value of $\theta$ for a given sine value.

Q: What is the inverse sine function?

A: The inverse sine function, denoted as $\sin^{-1}x$, is a function that returns the angle whose sine is equal to the given value. It is the inverse of the sine function.

Q: How do I use a calculator to find the inverse sine?

A: To find the inverse sine of a value using a calculator, you can enter the value into the calculator and press the inverse sine button, denoted as $\sin^{-1}$. Most scientific calculators have this button.

Q: What if the calculator does not have an inverse sine button?

A: If your calculator does not have an inverse sine button, you can use the arcsine function, which is the same as the inverse sine function. The arcsine function is denoted as $\arcsin x$.

Q: How do I find other possible values of $\theta$?

A: To find other possible values of $\theta$, you can use the fact that the sine function is periodic with a period of $360^{\circ}$. This means that if you add or subtract multiples of $360^{\circ}$ to the original angle, you will get other angles that have the same sine value.

Q: What if I want to find the value of $\theta$ in a different interval?

A: If you want to find the value of $\theta$ in a different interval, you can use the same method as before, but you will need to adjust the interval accordingly. For example, if you want to find the value of $\theta$ in the interval $\left[90^{\circ}, 180^{\circ}\right]$, you will need to add $180^{\circ}$ to the original angle.

Q: Can I use a calculator to find the value of $\theta$ in a different interval?

A: Yes, you can use a calculator to find the value of $\theta$ in a different interval. You can enter the value of the interval into the calculator and use the inverse sine function to find the value of $\theta$.

Q: What if I get a negative value for $\theta$?

A: If you get a negative value for $\theta$, it means that the angle is in the second or third quadrant. You can use the fact that the sine function is negative in the second and third quadrants to find the correct value of $\theta$.

Q: Can I use a calculator to find the value of $\theta$ in the second or third quadrant?

A: Yes, you can use a calculator to find the value of $\theta$ in the second or third quadrant. You can enter the value of the angle into the calculator and use the inverse sine function to find the value of $\theta$.

Conclusion

In this article, we answered some frequently asked questions related to finding the value of $\theta$ for a given sine value. We hope that this article has been helpful in clarifying any confusion you may have had.

Final Answer

The final answer is: $\boxed{76.703}$