Find The Value Of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right$\].A. $\sqrt{3}$B. $\frac{\sqrt{3}}{3}$C. $-\frac{\sqrt{3}}{3}$D. $-\sqrt{3}$

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Introduction

In this article, we will explore the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. This problem involves the use of trigonometric functions, specifically the tangent and sine functions. We will start by understanding the given expression and then proceed to simplify it step by step.

Understanding the Expression

The given expression is $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. This expression involves the inverse sine function, denoted by $\sin ^{-1}$. The inverse sine function returns the angle whose sine is a given value. In this case, we are given the value $-\frac{1}{2}$, and we need to find the angle whose sine is equal to this value.

Finding the Angle Whose Sine is $-\frac{1}{2}$

To find the angle whose sine is $-\frac{1}{2}$, we can use the unit circle or a trigonometric table. From the unit circle, we know that the sine of $-\frac{\pi}{6}$ is $-\frac{1}{2}$. Therefore, the angle whose sine is $-\frac{1}{2}$ is $-\frac{\pi}{6}$.

Substituting the Angle into the Expression

Now that we have found the angle whose sine is $-\frac{1}{2}$, we can substitute this angle into the original expression. The expression becomes $\tan \left(-\frac{\pi}{6}\right)$.

Evaluating the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions. Therefore, we can evaluate the tangent function by finding the ratio of the sine and cosine of the angle $-\frac{\pi}{6}$.

Finding the Sine and Cosine of $-\frac{\pi}{6}$

From the unit circle, we know that the sine of $-\frac{\pi}{6}$ is $-\frac{1}{2}$ and the cosine of $-\frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$. Therefore, the ratio of the sine and cosine of $-\frac{\pi}{6}$ is $\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$.

Simplifying the Ratio

We can simplify the ratio by multiplying the numerator and denominator by 2. This gives us $\frac{-1}{\sqrt{3}}$.

Rationalizing the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by $\sqrt{3}$. This gives us $\frac{-\sqrt{3}}{3}$.

Conclusion

In conclusion, the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$ is $\frac{-\sqrt{3}}{3}$. This is the final answer to the problem.

Final Answer

The final answer is $\boxed{\frac{-\sqrt{3}}{3}}$.

Discussion

This problem involves the use of trigonometric functions, specifically the tangent and sine functions. The inverse sine function is used to find the angle whose sine is a given value. The tangent function is then used to evaluate the expression. The final answer is $\frac{-\sqrt{3}}{3}$.

Related Problems

• Find the value of $\tan \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$.
• Find the value of $\sin \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$.
• Find the value of $\cos \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)$.

Solved Problems

• Find the value of $\tan \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)$.
• Find the value of $\sin \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$.
• Find the value of $\cos \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$.

Practice Problems

• Find the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$.
• Find the value of $\sin \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$.
• Find the value of $\cos \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$.

Solutions

• Find the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$: $\frac{-\sqrt{3}}{3}$.
• Find the value of $\sin \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$: $\frac{\sqrt{3}}{2}$.
• Find the value of $\cos \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$: $\frac{1}{2}$.

Conclusion

In this article, we have explored the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. We have used the inverse sine function to find the angle whose sine is $-\frac{1}{2}$ and then evaluated the tangent function to find the final answer. The final answer is $\frac{-\sqrt{3}}{3}$.

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Introduction

In our previous article, we explored the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. We used the inverse sine function to find the angle whose sine is $-\frac{1}{2}$ and then evaluated the tangent function to find the final answer. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the inverse sine function?

A: The inverse sine function, denoted by $\sin ^{-1}$, is a function that returns the angle whose sine is a given value. In other words, if we know the sine of an angle, we can use the inverse sine function to find the angle itself.

Q: How do we find the angle whose sine is $-\frac{1}{2}$?

A: To find the angle whose sine is $-\frac{1}{2}$, we can use the unit circle or a trigonometric table. From the unit circle, we know that the sine of $-\frac{\pi}{6}$ is $-\frac{1}{2}$. Therefore, the angle whose sine is $-\frac{1}{2}$ is $-\frac{\pi}{6}$.

Q: What is the tangent function?

A: The tangent function, denoted by $\tan$, is a function that returns the ratio of the sine and cosine of an angle. In other words, if we know the sine and cosine of an angle, we can use the tangent function to find the ratio of these two values.

Q: How do we evaluate the tangent function?

A: To evaluate the tangent function, we need to find the ratio of the sine and cosine of the angle. In this case, we need to find the ratio of the sine and cosine of $-\frac{\pi}{6}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\frac{-\sqrt{3}}{3}$.

Q: Can you explain the solution in more detail?

A: Of course! The solution involves using the inverse sine function to find the angle whose sine is $-\frac{1}{2}$, and then evaluating the tangent function to find the ratio of the sine and cosine of this angle. We can then simplify the ratio to find the final answer.

Q: What are some related problems that we can solve?

A: Some related problems that we can solve include finding the value of $\tan \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$, finding the value of $\sin \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$, and finding the value of $\cos \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)$.

Conclusion

In this article, we have answered some frequently asked questions related to the problem of finding the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. We have explained the solution in more detail and provided some related problems that we can solve.

Final Answer

The final answer to the problem is $\boxed{\frac{-\sqrt{3}}{3}}$.

Related Articles

• Find the Value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$
• Find the Value of $\tan \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$
• Find the Value of $\sin \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$

Practice Problems

• Find the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$.
• Find the value of $\sin \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$.
• Find the value of $\cos \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$.

Solutions

• Find the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$: $\frac{-\sqrt{3}}{3}$.
• Find the value of $\sin \left(\cos ^{-1}\left(\frac{1}{2}\right)\right)$: $\frac{\sqrt{3}}{2}$.
• Find the value of $\cos \left(\tan ^{-1}\left(\sqrt{3}\right)\right)$: $\frac{1}{2}$.

Conclusion

In this article, we have provided some frequently asked questions and answers related to the problem of finding the value of $\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$. We have explained the solution in more detail and provided some related problems that we can solve.