Find A Value Of $\theta$ In The Interval $\left[0^{\circ}, 90^{\circ}\right\] That Satisfies The Given Statement.$\sin \theta = 0.83485668$\theta \approx \square^{\circ}$(Simplify Your Answer. Type An Integer Or A

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 identities and inverse trigonometric functions to find the corresponding angle. In this article, we will explore how to find the value of $\theta$ in the interval $\left[0^{\circ}, 90^{\circ}\right]$ that satisfies the given statement $\sin \theta = 0.83485668$.

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 Value of $\theta$

To find the value of $\theta$ that satisfies the given statement, we can use the inverse sine function, denoted as $\sin^{-1}$. 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 0.83485668.

Using a Calculator to Find the Value of $\theta$

We can use a calculator to find the value of $\theta$ by entering the sine value and using the inverse sine function. Most calculators have a button labeled as "sin^-1" or "arcsin" that we can use to find the inverse sine.

Calculating the Value of $\theta$

Using a calculator, we can find the value of $\theta$ as follows:

$\theta \approx \sin^{-1}(0.83485668)$

$\theta \approx 56.04^{\circ}$

Rounding the Value of $\theta$

Since we are looking for the value of $\theta$ in the interval $\left[0^{\circ}, 90^{\circ}\right]$, we can round the value of $\theta$ to the nearest integer.

Conclusion

In conclusion, we have found the value of $\theta$ that satisfies the given statement $\sin \theta = 0.83485668$. The value of $\theta$ is approximately $56^{\circ}$.

Additional Information

It's worth noting that the sine function is periodic, meaning that it repeats itself every $360^{\circ}$. Therefore, there are multiple angles that satisfy the given statement, but we are only interested in the value of $\theta$ in the interval $\left[0^{\circ}, 90^{\circ}\right]$.

Final Answer

The final answer is: $\boxed{56}$

Introduction

In our previous article, we explored how to find the value of $\theta$ in the interval $\left[0^{\circ}, 90^{\circ}\right]$ that satisfies the given statement $\sin \theta = 0.83485668$. In this article, we will answer some frequently asked questions related to finding the value of $\theta$ for a given sine value.

Q&A

Q: What is the inverse sine function?

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

Q: How do I use a calculator to find the value of $\theta$?

A: To use a calculator to find the value of $\theta$, you need to enter the sine value and use the inverse sine function. Most calculators have a button labeled as "sin^-1" or "arcsin" that you can use to find the inverse sine.

Q: What if the sine value is not in the interval $\left[-1, 1\right]$?

A: If the sine value is not in the interval $\left[-1, 1\right]$, you need to adjust the value to be within the interval. You can do this by adding or subtracting multiples of $2\pi$ to the value.

Q: Can I use a calculator to find the value of $\theta$ for a given sine value?

A: Yes, you can use a calculator to find the value of $\theta$ for a given sine value. Most calculators have a button labeled as "sin^-1" or "arcsin" that you can use to find the inverse sine.

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 adjust the value to be within the interval $\left[0^{\circ}, 90^{\circ}\right]$ by adding $180^{\circ}$ to the value.

Q: Can I use a calculator to find the value of $\theta$ for a given sine value in degrees?

A: Yes, you can use a calculator to find the value of $\theta$ for a given sine value in degrees. Most calculators have a button labeled as "sin^-1" or "arcsin" that you can use to find the inverse sine.

Q: What if I get a value for $\theta$ that is not in the interval $\left[0^{\circ}, 90^{\circ}\right]$?

A: If you get a value for $\theta$ that is not in the interval $\left[0^{\circ}, 90^{\circ}\right]$, it means that the angle is in the second or third quadrant. You can adjust the value to be within the interval $\left[0^{\circ}, 90^{\circ}\right]$ by adding $180^{\circ}$ to the value.

Conclusion

In conclusion, we have 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 doubts you may have had.

Additional Information

It's worth noting that the sine function is periodic, meaning that it repeats itself every $360^{\circ}$. Therefore, there are multiple angles that satisfy the given statement, but we are only interested in the value of $\theta$ in the interval $\left[0^{\circ}, 90^{\circ}\right]$.

Final Answer

The final answer is: $\boxed{56}$