Suppose $x=\sin ^{-1}\left(\frac{3}{14}\right)$.Part 1:- Then $x$ Will Be An Angle In Quadrant $\square$ 1. (List All That Apply)- $\sin (x)=$ $\square$Part 2 Of 4:

Introduction

The inverse sine function, denoted as $\sin^{-1}(x)$, is a mathematical operation that returns the angle whose sine is a given value. In this article, we will explore the concept of the inverse sine function and its application in identifying angles in different quadrants. We will also examine the properties of the inverse sine function and its relationship with the sine function.

The Inverse Sine Function

The inverse sine function is defined as the inverse of the sine function. It is denoted as $\sin^{-1}(x)$ and is defined as the angle whose sine is equal to $x$. In other words, if $\sin(y) = x$, then $y = \sin^{-1}(x)$. The range of the inverse sine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which corresponds to the angles in the second and fourth quadrants.

Quadrant Identification

Quadrants are regions on the Cartesian plane that are defined by the x-axis and y-axis. The four quadrants are:

• Quadrant I: $(0, \frac{\pi}{2})$
• Quadrant II: $(-\frac{\pi}{2}, 0)$
• Quadrant III: $(0, \frac{\pi}{2})$
• Quadrant IV: $(-\frac{\pi}{2}, 0)$

To identify the quadrant in which an angle lies, we can use the following rules:

• If the angle is in Quadrant I, then $\sin(x) > 0$ and $\cos(x) > 0$.
• If the angle is in Quadrant II, then $\sin(x) < 0$ and $\cos(x) > 0$.
• If the angle is in Quadrant III, then $\sin(x) < 0$ and $\cos(x) < 0$.
• If the angle is in Quadrant IV, then $\sin(x) > 0$ and $\cos(x) < 0$.

Part 1: Identifying the Quadrant

Given that $x = \sin^{-1}\left(\frac{3}{14}\right)$, we need to determine the quadrant in which $x$ lies. Since the inverse sine function returns an angle in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$, we know that $x$ lies in either Quadrant II or Quadrant IV.

To determine the quadrant, we can use the following steps:

1. Evaluate the sine function at $x$: $\sin(x) = \frac{3}{14}$.
2. Determine the sign of the sine function: Since $\sin(x) > 0$, we know that $x$ lies in either Quadrant I or Quadrant IV.
3. Determine the sign of the cosine function: Since $\cos(x) > 0$ for angles in Quadrant I and $\cos(x) < 0$ for angles in Quadrant IV, we can conclude that $x$ lies in Quadrant I.

Therefore, the correct answer is:

• Quadrant I

Part 2: Evaluating the Sine Function

Now that we have identified the quadrant in which $x$ lies, we can evaluate the sine function at $x$. Since $\sin(x) = \frac{3}{14}$, we can conclude that:

• $\sin(x) = \frac{3}{14}$

Conclusion

In this article, we have explored the concept of the inverse sine function and its application in identifying angles in different quadrants. We have also examined the properties of the inverse sine function and its relationship with the sine function. By following the steps outlined in this article, we can determine the quadrant in which an angle lies and evaluate the sine function at that angle.

References

• [1] "Inverse Sine Function" by MathWorld
• [2] "Quadrant Identification" by Khan Academy

Further Reading

• "Inverse Trigonometric Functions" by Wolfram MathWorld
• "Quadrant Identification" by MIT OpenCourseWare
  Inverse Sine Function and Quadrant Identification: Q&A =====================================================

Introduction

In our previous article, we explored the concept of the inverse sine function and its application in identifying angles in different quadrants. In this article, we will answer some frequently asked questions related to the inverse sine function and quadrant identification.

Q&A

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 $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which corresponds to the angles in the second and fourth quadrants.

Q: How do I determine the quadrant in which an angle lies?

A: To determine the quadrant in which an angle lies, you can use the following rules:

• If the angle is in Quadrant I, then $\sin(x) > 0$ and $\cos(x) > 0$.
• If the angle is in Quadrant II, then $\sin(x) < 0$ and $\cos(x) > 0$.
• If the angle is in Quadrant III, then $\sin(x) < 0$ and $\cos(x) < 0$.
• If the angle is in Quadrant IV, then $\sin(x) > 0$ and $\cos(x) < 0$.

Q: How do I evaluate the sine function at an angle?

A: To evaluate the sine function at an angle, you can use the following steps:

1. Evaluate the sine function at the angle: $\sin(x)$.
2. Determine the sign of the sine function: If $\sin(x) > 0$, then the angle lies in Quadrant I or Quadrant IV. If $\sin(x) < 0$, then the angle lies in Quadrant II or Quadrant III.

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

A: The inverse sine function is the inverse of the sine function. In other words, if $\sin(y) = x$, then $y = \sin^{-1}(x)$.

Q: Can I use the inverse sine function to find the angle whose cosine is a given value?

A: No, the inverse sine function is only used to find the angle whose sine is a given value. To find the angle whose cosine is a given value, you can use the inverse cosine function.

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:

• Finding the angle of elevation or depression of an object.
• Determining the angle of a triangle.
• Calculating the area of a triangle.

Conclusion

In this article, we have answered some frequently asked questions related to the inverse sine function and quadrant identification. We hope that this article has provided you with a better understanding of the inverse sine function and its applications.

References

• [1] "Inverse Sine Function" by MathWorld
• [2] "Quadrant Identification" by Khan Academy

Further Reading

• "Inverse Trigonometric Functions" by Wolfram MathWorld
• "Quadrant Identification" by MIT OpenCourseWare