8. If $y=\sin ^{-1} X$, Then What Is The Range?A. $-\frac{\pi}{2} \leq Y \leq \frac{\pi}{2}$B. $0 \leq Y \leq \pi$C. $-\frac{\pi}{2} \ \textless \ Y \ \textless \ \frac{\pi}{2}$D. $0 \ \textless \ Y \

Introduction

The inverse sine function, denoted as $\sin^{-1} x$, is a mathematical function that returns the angle whose sine is a given number. In this article, we will explore the range of the inverse sine function, which is a crucial concept in mathematics, particularly in trigonometry and calculus.

What is the Inverse Sine Function?

The inverse sine function is a function that takes a real number $x$ as input and returns an angle $y$ such that $\sin y = x$. In other words, it is a function that returns the angle whose sine is equal to the input value. The inverse sine function is also known as the arcsine function.

Range of the Inverse Sine Function

To determine the range of the inverse sine function, we need to consider the possible values of the input $x$. The sine function has a range of $[-1, 1]$, which means that the input $x$ can take any value between $-1$ and $1$. However, the inverse sine function returns an angle $y$ such that $\sin y = x$. Since the sine function is periodic with a period of $2\pi$, the angle $y$ can take any value between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Why is the Range of the Inverse Sine Function $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$?

The range of the inverse sine function is $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ because the sine function is positive in the first and second quadrants of the unit circle. In these quadrants, the angle $y$ can take any value between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. When $x$ is positive, the angle $y$ is in the first quadrant, and when $x$ is negative, the angle $y$ is in the second quadrant.

Why is the Range of the Inverse Sine Function Not $0 \leq y \leq \pi$?

The range of the inverse sine function is not $0 \leq y \leq \pi$ because the sine function is not one-to-one in the interval $[0, \pi]$. In this interval, the sine function takes on the same value at two different angles, which means that the inverse sine function is not well-defined.

Why is the Range of the Inverse Sine Function Not $-\frac{\pi}{2} \ \textless \ y \ \textless \ \frac{\pi}{2}$?

The range of the inverse sine function is not $-\frac{\pi}{2} \ \textless \ y \ \textless \ \frac{\pi}{2}$ because the angle $y$ can take on the value $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, which are the endpoints of the interval.

Why is the Range of the Inverse Sine Function Not $0 \ \textless \ y \ \textless \ \pi$?

The range of the inverse sine function is not $0 \ \textless \ y \ \textless \ \pi$ because the angle $y$ can take on the value $0$, which is not in the interval.

Conclusion

In conclusion, the range of the inverse sine function is $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$. This is because the sine function is positive in the first and second quadrants of the unit circle, and the angle $y$ can take any value between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. The other options are incorrect because they do not accurately reflect the range of the inverse sine function.

Final Answer

The final answer is A. $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.

Introduction

The inverse sine function, denoted as $\sin^{-1} x$, is a mathematical function that returns the angle whose sine is a given number. In this article, we will provide a comprehensive Q&A guide to help you understand the inverse sine function and its applications.

Q: What is the inverse sine function?

A: The inverse sine function is a function that takes a real number $x$ as input and returns an angle $y$ such that $\sin y = x$. In other words, it is a function that returns the angle whose sine is equal to the input value.

Q: What is the range of the inverse sine function?

A: The range of the inverse sine function is $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$. This is because the sine function is positive in the first and second quadrants of the unit circle, and the angle $y$ can take any value between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Q: Why is the range of the inverse sine function not $0 \leq y \leq \pi$?

A: The range of the inverse sine function is not $0 \leq y \leq \pi$ because the sine function is not one-to-one in the interval $[0, \pi]$. In this interval, the sine function takes on the same value at two different angles, which means that the inverse sine function is not well-defined.

Q: What is the domain of the inverse sine function?

A: The domain of the inverse sine function is $[-1, 1]$. This is because the sine function has a range of $[-1, 1]$, and the inverse sine function returns an angle $y$ such that $\sin y = x$.

Q: How do I evaluate the inverse sine function?

A: To evaluate the inverse sine function, you can use a calculator or a computer program that has a built-in inverse sine function. Alternatively, you can use the following formula:

$$\sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x$$

Q: What are some common applications of the inverse sine function?

A: The inverse sine function has many applications in mathematics, physics, and engineering. Some common applications include:

• Calculating the angle of elevation or depression of an object
• Finding the area of a triangle or a sector of a circle
• Determining the length of a side of a triangle or a chord of a circle
• Solving problems involving right triangles or circular arcs

Q: Can I use the inverse sine function to solve problems involving complex numbers?

A: Yes, you can use the inverse sine function to solve problems involving complex numbers. However, you need to be careful when working with complex numbers, as the inverse sine function may return a complex value.

Q: What are some common mistakes to avoid when working with the inverse sine function?

A: Some common mistakes to avoid when working with the inverse sine function include:

• Not checking the domain of the function before evaluating it
• Not using the correct formula for the inverse sine function
• Not considering the range of the function when solving problems
• Not using a calculator or computer program to evaluate the function when necessary

Conclusion

In conclusion, the inverse sine function is a powerful mathematical tool that has many applications in mathematics, physics, and engineering. By understanding the range, domain, and applications of the inverse sine function, you can solve a wide range of problems involving right triangles, circular arcs, and complex numbers.

Final Tips

• Always check the domain of the inverse sine function before evaluating it.
• Use the correct formula for the inverse sine function.
• Consider the range of the function when solving problems.
• Use a calculator or computer program to evaluate the function when necessary.

Additional Resources

• For more information on the inverse sine function, see the following resources:
  • Khan Academy: Inverse Sine Function
  • Mathway: Inverse Sine Function
  • Wolfram Alpha: Inverse Sine Function