Understanding the Inverse Sine Function Graph

Oct 13, 2025 by windailys.com 48 views

The inverse sine function, denoted as $\sin^$ , is a mathematical operation that returns the angle whose sine is a given value. In this article, we will explore the equivalent value of $\sin^{-1(-0.4)$ and round our answer to the nearest hundredth of a radian.

What is the Inverse Sine Function?

The inverse sine function is a trigonometric function that is the inverse of the sine function. It is defined as the angle whose sine is a given value. The sine function is a periodic function that oscillates between -1 and 1, and the inverse sine function returns an angle in the range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ .

Understanding the Inverse Sine Function Graph

The graph of the inverse sine function is a reflection of the graph of the sine function about the line $y = x$ . The graph of the inverse sine function is a curve that approaches the line $y = -\frac{\pi}{2}$ as $x$ approaches -1, and approaches the line $y = \frac{\pi}{2}$ as $x$ approaches 1.

Finding the Equivalent Value of $\sin^{-1}(-0.4)$

To find the equivalent value of $\sin^{-1}(-0.4)$ , we need to find the angle whose sine is -0.4. We can use a calculator or a trigonometric table to find the value of $\sin^{-1}(-0.4)$ .

Using a Calculator to Find the Equivalent Value

Using a calculator, we can find the value of $\sin^{-1}(-0.4)$ by entering the value -0.4 into the calculator and pressing the inverse sine button. The calculator will return the value of $\sin^{-1}(-0.4)$ in radians.

Rounding the Answer to the Nearest Hundredth of a Radian

Once we have found the value of $\sin^{-1}(-0.4)$ , we need to round our answer to the nearest hundredth of a radian. This means that we need to round the value to two decimal places.

Evaluating the Answer Choices

Now that we have found the value of $\sin^{-1}(-0.4)$ , we can evaluate the answer choices to see which one is equivalent to our answer.

A. $-2.50$
B. $-0.60$
C. $-0.41$
D. $-0.39$

Conclusion

In conclusion, the equivalent value of $\sin^{-1}(-0.4)$ is $-0.41$ radians, rounded to the nearest hundredth of a radian. This is the correct answer choice.

Final Answer

In this article, we will answer some frequently asked questions about the inverse sine function.

Q: What is the inverse sine function?

A: The inverse sine function, denoted as $\sin^{-1}(x)$ , is a mathematical operation that returns the angle whose sine is a given value.

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

A: The range of the inverse sine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$ .

Q: How do I find the equivalent value of $\sin^{-1}(x)$ ?

A: To find the equivalent value of $\sin^{-1}(x)$ , you can use a calculator or a trigonometric table to find the value of $\sin^{-1}(x)$ .

Q: What is the difference between the sine function and the inverse sine function?

A: The sine function is a periodic function that oscillates between -1 and 1, while the inverse sine function returns an angle in the range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ .

Q: Can I use the inverse sine function to find the angle of a right triangle?

A: Yes, you can use the inverse sine function to find the angle of a right triangle. If you know the length of the opposite side and the hypotenuse, you can use the inverse sine function to find the angle.

Q: How do I round my answer to the nearest hundredth of a radian?

A: To round your answer to the nearest hundredth of a radian, you need to round the value to two decimal places.

Q: What is the equivalent value of $\sin^{-1}(-0.4)$ ?

A: The equivalent value of $\sin^{-1}(-0.4)$ is $-0.41$ radians, rounded to the nearest hundredth of a radian.

Q: Can I use the inverse sine function to find the value of $\sin(x)$ ?

A: No, you cannot use the inverse sine function to find the value of $\sin(x)$ . The inverse sine function returns the angle whose sine is a given value, not the value of $\sin(x)$ .

Q: What is the inverse sine function used for in real-world applications?

A: The inverse sine function is used in a variety of real-world applications, including:

Calculating the angle of a right triangle
Finding the value of $\sin(x)$
Solving trigonometric equations
Calculating the area of a triangle

Conclusion

In conclusion, the inverse sine function is a powerful mathematical tool that can be used to find the angle of a right triangle, solve trigonometric equations, and calculate the area of a triangle. By understanding the inverse sine function and how to use it, you can solve a variety of mathematical problems and apply the concept to real-world applications.

Final Answer

The final answer is $\boxed{-0.41}$ .