Find The Exact Value Of \tan^{-1}\left(\sin \frac{3\pi}{2}\right ].

Introduction

In this article, we will delve into the world of trigonometry and explore the concept of inverse trigonometric functions. Specifically, we will focus on finding the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$. This problem may seem daunting at first, but with a step-by-step approach and a solid understanding of trigonometric concepts, we can arrive at a precise solution.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are a set of functions that return the angle whose trigonometric function satisfies a given equation. For example, the inverse sine function, denoted as $\sin^{-1}(x)$, returns the angle whose sine is equal to $x$. Similarly, the inverse tangent function, denoted as $\tan^{-1}(x)$, returns the angle whose tangent is equal to $x$.

Evaluating $\sin \frac{3\pi}{2}$

To find the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$, we first need to evaluate $\sin \frac{3\pi}{2}$. Using the unit circle or trigonometric identities, we can determine that $\sin \frac{3\pi}{2} = -1$.

Understanding the Range of Inverse Tangent Function

The range of the inverse tangent function, $\tan^{-1}(x)$, is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. This means that the output of the inverse tangent function will always be an angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Finding the Exact Value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$

Now that we have evaluated $\sin \frac{3\pi}{2}$ and understand the range of the inverse tangent function, we can find the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$. Since $\sin \frac{3\pi}{2} = -1$, we need to find the angle whose tangent is equal to $-1$.

Using Trigonometric Identities to Find the Angle

Using trigonometric identities, we can determine that the angle whose tangent is equal to $-1$ is $\frac{3\pi}{4}$. However, this angle is outside the range of the inverse tangent function, which is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Finding the Equivalent Angle within the Range

To find the equivalent angle within the range of the inverse tangent function, we can use the fact that the tangent function is periodic with a period of $\pi$. This means that we can add or subtract multiples of $\pi$ to the angle $\frac{3\pi}{4}$ to find an equivalent angle within the range.

Finding the Final Answer

After adding or subtracting multiples of $\pi$, we can find the equivalent angle within the range of the inverse tangent function. In this case, we can add $-\pi$ to the angle $\frac{3\pi}{4}$ to get the equivalent angle $\frac{3\pi}{4} - \pi = -\frac{\pi}{4}$.

Conclusion

In conclusion, we have found the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$ by evaluating $\sin \frac{3\pi}{2}$, understanding the range of the inverse tangent function, and using trigonometric identities to find the equivalent angle within the range. The final answer is $\boxed{-\frac{\pi}{4}}$.

Additional Examples and Applications

Inverse trigonometric functions have numerous applications in mathematics, physics, and engineering. Some additional examples and applications include:

• Finding the area of a triangle using the inverse sine function
• Determining the length of a side of a right triangle using the inverse tangent function
• Calculating the height of a building using the inverse cosine function
• Modeling population growth using the inverse tangent function

Final Thoughts

In this article, we have explored the concept of inverse trigonometric functions and found the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$. We have used trigonometric identities, the unit circle, and the range of the inverse tangent function to arrive at a precise solution. This problem serves as a reminder of the importance of understanding trigonometric concepts and the power of inverse trigonometric functions in solving real-world problems.

Introduction

In our previous article, we explored the concept of inverse trigonometric functions and found the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$. In this article, we will answer some frequently asked questions related to this topic.

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

A: The range of the inverse tangent function, $\tan^{-1}(x)$, is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. This means that the output of the inverse tangent function will always be an angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Q: How do I evaluate $\sin \frac{3\pi}{2}$?

A: To evaluate $\sin \frac{3\pi}{2}$, you can use the unit circle or trigonometric identities. Using the unit circle, you can determine that $\sin \frac{3\pi}{2} = -1$.

Q: Why do I need to find the equivalent angle within the range of the inverse tangent function?

A: The inverse tangent function has a range of $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. If the angle you are trying to find is outside this range, you need to find the equivalent angle within the range.

Q: How do I find the equivalent angle within the range of the inverse tangent function?

A: To find the equivalent angle within the range of the inverse tangent function, you can use the fact that the tangent function is periodic with a period of $\pi$. This means that you can add or subtract multiples of $\pi$ to the angle to find an equivalent angle within the range.

Q: What are some additional examples and applications of inverse trigonometric functions?

A: Inverse trigonometric functions have numerous applications in mathematics, physics, and engineering. Some additional examples and applications include:

• Finding the area of a triangle using the inverse sine function
• Determining the length of a side of a right triangle using the inverse tangent function
• Calculating the height of a building using the inverse cosine function
• Modeling population growth using the inverse tangent function

Q: How do I use inverse trigonometric functions in real-world problems?

A: Inverse trigonometric functions can be used to solve a wide range of real-world problems. Some examples include:

• Calculating the height of a building or a mountain
• Determining the length of a side of a right triangle
• Finding the area of a triangle or a polygon
• Modeling population growth or decline

Q: What are some common mistakes to avoid when working with inverse trigonometric functions?

A: Some common mistakes to avoid when working with inverse trigonometric functions include:

• Not checking the range of the inverse tangent function
• Not using the correct trigonometric identity
• Not simplifying the expression before evaluating it
• Not checking the units of the answer

Q: How do I simplify expressions involving inverse trigonometric functions?

A: To simplify expressions involving inverse trigonometric functions, you can use trigonometric identities and algebraic manipulations. Some common techniques include:

• Using the sum and difference formulas for sine and cosine
• Using the double-angle and half-angle formulas for sine and cosine
• Simplifying the expression by combining like terms
• Using algebraic manipulations to isolate the inverse trigonometric function

Conclusion

In this article, we have answered some frequently asked questions related to finding the exact value of $\tan^{-1}\left(\sin \frac{3\pi}{2}\right)$. We have covered topics such as the range of the inverse tangent function, evaluating $\sin \frac{3\pi}{2}$, finding the equivalent angle within the range, and simplifying expressions involving inverse trigonometric functions. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of inverse trigonometric functions.