Suppose $x=\sin ^{-1}\left(\frac{3}{14}\right)$.Then $x$ Will Be An Angle In Quadrant(s) [ ] (list All That Apply).

Introduction

In trigonometry, the inverse sine function, denoted by $\sin^{-1}$, is used to find the angle whose sine is a given value. In this problem, we are given that $x=\sin^{-1}\left(\frac{3}{14}\right)$, and we need to determine the quadrant(s) in which the angle $x$ lies.

Understanding the Inverse Sine Function

The inverse sine function, $\sin^{-1}$, is defined as the angle whose sine is a given value. In other words, if $\sin(x) = y$, then $x = \sin^{-1}(y)$. The range of the inverse sine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which corresponds to the interval $[-90^\circ, 90^\circ]$.

Analyzing the Given Value

We are given that $x=\sin^{-1}\left(\frac{3}{14}\right)$. To determine the quadrant(s) in which the angle $x$ lies, we need to analyze the given value $\frac{3}{14}$. Since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants.

Determining the Quadrant(s)

To determine the specific quadrant(s) in which the angle $x$ lies, we need to consider the sign of the sine function. Since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants.

However, we need to consider the fact that the given value $\frac{3}{14}$ is positive. This means that the angle $x$ lies in the first quadrant, where the sine function is positive.

Conclusion

In conclusion, the angle $x$ lies in the first quadrant. Therefore, the correct answer is:

• Quadrant I

Additional Considerations

It's worth noting that the angle $x$ may also lie in the second quadrant, depending on the value of the given expression $\sin^{-1}\left(\frac{3}{14}\right)$. However, since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants.

Final Answer

The final answer is:

• Quadrant I

Explanation

The explanation for the final answer is as follows:

Since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants. However, since the given value $\frac{3}{14}$ is positive, we can conclude that the angle $x$ lies in the first quadrant.

Additional Information

It's worth noting that the angle $x$ may also lie in the second quadrant, depending on the value of the given expression $\sin^{-1}\left(\frac{3}{14}\right)$. However, since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Glossary

• Inverse Sine Function: The inverse sine function, denoted by $\sin^{-1}$, is used to find the angle whose sine is a given value.
• Quadrant: A quadrant is a region of the coordinate plane, defined by the intersection of two axes.
• Sine Function: The sine function, denoted by $\sin$, is a trigonometric function that relates the ratio of the length of the side opposite a given angle to the length of the hypotenuse of a right triangle.

FAQs

• Q: What is the inverse sine function? A: The inverse sine function, denoted by $\sin^{-1}$, is used to find 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 interval $[-90^\circ, 90^\circ]$.
• Q: What is the sign of the sine function in the first and second quadrants? A: The sine function is positive in the first and second quadrants.
  

Q&A: Inverse Sine Function and Quadrants

Q: What is the inverse sine function?

A: The inverse sine function, denoted by $\sin^{-1}$, is used to find 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 interval $[-90^\circ, 90^\circ]$.

Q: What is the sign of the sine function in the first and second quadrants?

A: The sine function is positive in the first and second quadrants.

Q: Can the angle $x$ lie in the third or fourth quadrant?

A: No, the angle $x$ cannot lie in the third or fourth quadrant. Since the sine function is positive in the first and second quadrants, the angle $x$ must lie in one of these two quadrants.

Q: How can we determine the specific quadrant(s) in which the angle $x$ lies?

A: To determine the specific quadrant(s) in which the angle $x$ lies, we need to consider the sign of the sine function. Since the sine function is positive in the first and second quadrants, we can conclude that the angle $x$ lies in one of these two quadrants.

Q: What is the value of the given expression $\sin^{-1}\left(\frac{3}{14}\right)$?

A: The value of the given expression $\sin^{-1}\left(\frac{3}{14}\right)$ is an angle in the first quadrant.

Q: Can the angle $x$ lie in the second quadrant?

A: Yes, the angle $x$ can lie in the second quadrant. Since the sine function is positive in the first and second quadrants, the angle $x$ can lie in either of these two quadrants.

Q: How can we determine the exact quadrant(s) in which the angle $x$ lies?

A: To determine the exact quadrant(s) in which the angle $x$ lies, we need to consider the value of the given expression $\sin^{-1}\left(\frac{3}{14}\right)$. If the value is positive, then the angle $x$ lies in the first quadrant. If the value is negative, then the angle $x$ lies in the second quadrant.

Q: What is the final answer?

A: The final answer is Quadrant I.

Q: Can the angle $x$ lie in other quadrants?

A: No, the angle $x$ cannot lie in other quadrants. Since the sine function is positive in the first and second quadrants, the angle $x$ must lie in one of these two quadrants.

Q: What is the significance of the inverse sine function in trigonometry?

A: The inverse sine function is used to find the angle whose sine is a given value. It is an important concept in trigonometry and is used to solve problems involving right triangles.

Q: How can we use the inverse sine function to solve problems?

A: We can use the inverse sine function to solve problems involving right triangles by finding the angle whose sine is a given value. This can be done using a calculator or by using the inverse sine function formula.

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

A: The inverse sine function has many common applications in trigonometry, including solving problems involving right triangles, finding the angle of a given sine value, and determining the quadrant(s) in which an angle lies.

Q: Can the inverse sine function be used to solve problems involving other trigonometric functions?

A: Yes, the inverse sine function can be used to solve problems involving other trigonometric functions, such as the cosine and tangent functions. However, the inverse sine function is specifically used to find the angle whose sine is a given value.

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

A: The inverse sine function and the sine function are related in that the inverse sine function is used to find the angle whose sine is a given value. In other words, if $\sin(x) = y$, then $x = \sin^{-1}(y)$.

Q: Can the inverse sine function be used to solve problems involving complex numbers?

A: Yes, the inverse sine function can be used to solve problems involving complex numbers. However, the inverse sine function is typically used to find the angle whose sine is a given value, and complex numbers are typically used to represent points in the complex plane.

Q: What is the significance of the inverse sine function in calculus?

A: The inverse sine function is used in calculus to find the area under curves and to solve problems involving integration. It is an important concept in calculus and is used to solve problems involving right triangles and other geometric shapes.

Q: Can the inverse sine function be used to solve problems involving other mathematical concepts?

A: Yes, the inverse sine function can be used to solve problems involving other mathematical concepts, such as algebra and geometry. However, the inverse sine function is specifically used to find the angle whose sine is a given value.

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

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

• Using the inverse sine function to solve problems involving other trigonometric functions
• Using the inverse sine function to find the angle whose cosine or tangent is a given value
• Using the inverse sine function to solve problems involving complex numbers
• Using the inverse sine function to find the area under curves or to solve problems involving integration

Q: How can we ensure that we are using the inverse sine function correctly?

A: To ensure that we are using the inverse sine function correctly, we need to follow the rules and guidelines for using the inverse sine function. This includes:

• Using the inverse sine function to find the angle whose sine is a given value
• Using the correct formula for the inverse sine function
• Checking the quadrant(s) in which the angle lies
• Using the inverse sine function to solve problems involving right triangles and other geometric shapes

Q: What are some common applications of the inverse sine function in real-world problems?

A: The inverse sine function has many common applications in real-world problems, including:

• Finding the angle of a given sine value in a right triangle
• Determining the quadrant(s) in which an angle lies
• Solving problems involving right triangles and other geometric shapes
• Finding the area under curves and solving problems involving integration

Q: Can the inverse sine function be used to solve problems involving other mathematical concepts in real-world problems?

A: Yes, the inverse sine function can be used to solve problems involving other mathematical concepts in real-world problems. However, the inverse sine function is specifically used to find the angle whose sine is a given value.

Q: What are some common challenges when using the inverse sine function in real-world problems?

A: Some common challenges when using the inverse sine function in real-world problems include:

• Finding the correct formula for the inverse sine function
• Checking the quadrant(s) in which the angle lies
• Using the inverse sine function to solve problems involving complex numbers
• Using the inverse sine function to find the area under curves or to solve problems involving integration

Q: How can we overcome these challenges when using the inverse sine function in real-world problems?

A: To overcome these challenges when using the inverse sine function in real-world problems, we need to:

• Follow the rules and guidelines for using the inverse sine function
• Use the correct formula for the inverse sine function
• Check the quadrant(s) in which the angle lies
• Use the inverse sine function to solve problems involving right triangles and other geometric shapes

Q: What are some common resources for learning more about the inverse sine function?

A: Some common resources for learning more about the inverse sine function include:

• Textbooks on trigonometry and calculus
• Online resources and tutorials
• Calculators and software programs
• Online communities and forums

Q: Can the inverse sine function be used to solve problems involving other mathematical concepts in real-world problems?

A: Yes, the inverse sine function can be used to solve problems involving other mathematical concepts in real-world problems. However, the inverse sine function is specifically used to find the angle whose sine is a given value.

Q: What are some common applications of the inverse sine function in science and engineering?

A: The inverse sine function has many common applications in science and engineering, including:

• Finding the angle of a given sine value in a right triangle
• Determining the quadrant(s) in which an angle lies
• Solving problems involving right triangles and other geometric shapes
• Finding the area under curves and solving problems involving integration

Q: Can the inverse sine function be used to solve problems involving other mathematical concepts in science and engineering?

A: Yes, the inverse sine function can be used to solve problems involving other mathematical concepts in science and engineering. However, the inverse sine function is specifically used to find the angle whose sine is a given value.

Q: What are some common challenges when using the inverse sine function in science and engineering?

A: Some common challenges when using the inverse sine function in science and engineering include:

• Finding