Find The Exact Value Of \sin^{-1}\left(-\frac{1}{2}\right ]. Sin ⁡ − 1 ( − 1 2 ) = \sin^{-1}\left(-\frac{1}{2}\right) = Sin − 1 ( − 2 1 ​ ) =

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Introduction

In mathematics, the inverse sine function, denoted by $\sin^{-1}$, is used to find the angle whose sine is a given value. The sine function is a fundamental concept in trigonometry, and its inverse is used to solve equations involving the sine function. In this article, we will explore the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$.

Understanding the Inverse Sine Function

The inverse sine function, $\sin^{-1}$, is a function that takes a value between -1 and 1 as input and returns the angle whose sine is that value. The range of the inverse sine function is typically restricted to the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ to ensure that the function is one-to-one and has an inverse.

Properties of the Sine Function

The sine function has several important properties that are useful when working with the inverse sine function. One of the most important properties is that the sine function is periodic with period $2\pi$, meaning that the value of the sine function repeats every $2\pi$ radians. This property is useful when working with the inverse sine function, as it allows us to restrict the range of the function to a single period.

Finding the Exact Value of $\sin^{-1}\left(-\frac{1}{2}\right)$

To find the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$, we need to find the angle whose sine is $-\frac{1}{2}$. We can use the unit circle to find this angle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.

Using the Unit Circle to Find the Angle

To find the angle whose sine is $-\frac{1}{2}$, we need to find the point on the unit circle that has a y-coordinate of $-\frac{1}{2}$. We can do this by drawing a line from the origin to the point on the unit circle that has a y-coordinate of $-\frac{1}{2}$.

Finding the Angle Using the Inverse Sine Function

Once we have found the point on the unit circle that has a y-coordinate of $-\frac{1}{2}$, we can use the inverse sine function to find the angle whose sine is $-\frac{1}{2}$. The inverse sine function takes the y-coordinate of the point on the unit circle as input and returns the angle whose sine is that value.

Calculating the Angle

To calculate the angle, we need to use the inverse sine function. The inverse sine function is defined as:

$$\sin^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(\sqrt{1-x^2})$$

We can use this formula to calculate the angle whose sine is $-\frac{1}{2}$.

Substituting the Value of $x$

Substituting the value of $x$ into the formula, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{2} - \sin^{-1}\left(\sqrt{1-\left(-\frac{1}{2}\right)^2}\right)$$

Simplifying the Expression

Simplifying the expression, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{2} - \sin^{-1}\left(\sqrt{1-\frac{1}{4}}\right)$$

Evaluating the Expression

Evaluating the expression, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{2} - \sin^{-1}\left(\sqrt{\frac{3}{4}}\right)$$

Using the Inverse Sine Function to Find the Angle

Using the inverse sine function to find the angle, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{2} - \sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$$

Evaluating the Inverse Sine Function

Evaluating the inverse sine function, we get:

$$\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$$

Substituting the Value of the Inverse Sine Function

Substituting the value of the inverse sine function into the expression, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{2} - \frac{\pi}{3}$$

Simplifying the Expression

Simplifying the expression, we get:

$$\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{6}$$

Conclusion

In conclusion, the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$ is $\frac{\pi}{6}$. This value can be found using the unit circle and the inverse sine function. The inverse sine function is a powerful tool for solving equations involving the sine function, and it is an essential concept in trigonometry.

Final Answer

The final answer is: $\boxed{\frac{\pi}{6}}$

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Introduction

In our previous article, we explored the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$. In this article, we will answer some frequently asked questions about finding the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$.

Q1: What is the inverse sine function?

A1: The inverse sine function, denoted by $\sin^{-1}$, is a function that takes a value between -1 and 1 as input and returns the angle whose sine is that value.

Q2: Why is the range of the inverse sine function restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$?

A2: The range of the inverse sine function is restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ to ensure that the function is one-to-one and has an inverse.

Q3: How do you find the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$?

A3: To find the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$, you need to find the angle whose sine is $-\frac{1}{2}$. You can use the unit circle to find this angle.

Q4: What is the unit circle?

A4: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.

Q5: How do you use the unit circle to find the angle whose sine is $-\frac{1}{2}$?

A5: To find the angle whose sine is $-\frac{1}{2}$, you need to find the point on the unit circle that has a y-coordinate of $-\frac{1}{2}$. You can do this by drawing a line from the origin to the point on the unit circle that has a y-coordinate of $-\frac{1}{2}$.

Q6: What is the inverse sine function formula?

A6: The inverse sine function formula is:

$$\sin^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(\sqrt{1-x^2})$$

Q7: How do you use the inverse sine function formula to find the angle whose sine is $-\frac{1}{2}$?

A7: To find the angle whose sine is $-\frac{1}{2}$, you need to substitute the value of $x$ into the formula and simplify the expression.

Q8: What is the final answer to the problem?

A8: The final answer to the problem is $\boxed{\frac{\pi}{6}}$.

Q9: Why is the final answer $\boxed{\frac{\pi}{6}}$?

A9: The final answer is $\boxed{\frac{\pi}{6}}$ because the inverse sine function formula simplifies to $\frac{\pi}{6}$ when the value of $x$ is substituted.

Q10: What is the significance of finding the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$?

A10: Finding the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$ is significant because it helps to understand the properties of the inverse sine function and how it can be used to solve equations involving the sine function.

Conclusion

In conclusion, finding the exact value of $\sin^{-1}\left(-\frac{1}{2}\right)$ is an important concept in trigonometry. By understanding the properties of the inverse sine function and how it can be used to solve equations involving the sine function, you can gain a deeper understanding of the subject and improve your problem-solving skills.

Final Answer

The final answer is: $\boxed{\frac{\pi}{6}}$