Find The Exact Value Of The Following Expression, If Possible. Do Not Use A Calculator. \sin^{-1}\left(\sin \frac{3 \pi}{2}\right ]Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A.

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Introduction

In this article, we will explore the concept of inverse trigonometric functions and how to find the exact value of a given expression involving the inverse sine function. We will use mathematical concepts and properties to simplify the expression and arrive at the final answer.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle whose trigonometric function is a given value. For example, the inverse sine function, denoted by $\sin^{-1}x$, gives the angle whose sine is $x$. In other words, if $\sin \theta = x$, then $\sin^{-1}x = \theta$.

Evaluating the Expression

The given expression is $\sin^{-1}\left(\sin \frac{3 \pi}{2}\right)$. To evaluate this expression, we need to find the angle whose sine is equal to $\sin \frac{3 \pi}{2}$.

Properties of Sine Function

The sine function has a periodic nature, meaning that it repeats itself after a certain interval. The sine function has a period of $2 \pi$, which means that the value of the sine function at any angle is equal to the value of the sine function at that angle plus or minus any multiple of $2 \pi$.

Simplifying the Expression

Using the property of the sine function, we can simplify the expression as follows:

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

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

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

$= \sin^{-1}\left(-1\right)$

Finding the Angle

Now, we need to find the angle whose sine is equal to $-1$. We know that the sine function is negative in the third and fourth quadrants. Therefore, the angle whose sine is equal to $-1$ is in the third or fourth quadrant.

Using the Unit Circle

To find the angle, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin. The sine function is equal to the y-coordinate of the point on the unit circle corresponding to the angle.

Finding the Angle on the Unit Circle

Using the unit circle, we can find the angle whose sine is equal to $-1$. The point on the unit circle corresponding to the angle $\frac{3 \pi}{2}$ is $(0, -1)$. Therefore, the angle whose sine is equal to $-1$ is $\frac{3 \pi}{2}$.

Conclusion

In conclusion, the exact value of the expression $\sin^{-1}\left(\sin \frac{3 \pi}{2}\right)$ is $\frac{3 \pi}{2}$. This is because the sine function is equal to $-1$ at the angle $\frac{3 \pi}{2}$.

Final Answer

The final answer is: $\boxed{\frac{3 \pi}{2}}$

Introduction

In the previous article, we explored the concept of inverse trigonometric functions and how to find the exact value of a given expression involving the inverse sine function. In this article, we will answer some frequently asked questions related to the inverse sine function and its applications.

Q&A

Q1: What is the inverse sine function?

A1: The inverse sine function, denoted by $\sin^{-1}x$, gives the angle whose sine is $x$. In other words, if $\sin \theta = x$, then $\sin^{-1}x = \theta$.

Q2: What is the range of the inverse sine function?

A2: The range of the inverse sine function is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

Q3: How do I evaluate the expression $\sin^{-1}\left(\sin \frac{3 \pi}{2}\right)$?

A3: To evaluate the expression $\sin^{-1}\left(\sin \frac{3 \pi}{2}\right)$, we need to find the angle whose sine is equal to $\sin \frac{3 \pi}{2}$. Using the property of the sine function, we can simplify the expression as follows:

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

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

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

$= \sin^{-1}\left(-1\right)$

$= -\frac{\pi}{2}$

Q4: What is the difference between the sine function and the inverse sine function?

A4: The sine function gives the sine of an angle, while the inverse sine function gives the angle whose sine is a given value.

Q5: How do I use the inverse sine function in real-world applications?

A5: The inverse sine function has many real-world applications, such as:

• Calculating the angle of elevation of a building or a mountain
• Finding the angle of depression of a projectile
• Determining the angle of a triangle given the lengths of two sides
• Calculating the angle of a rotating object

Q6: Can I use the inverse sine function to find the angle of a right triangle?

A6: Yes, you can use the inverse sine function to find the angle of a right triangle. If you know the length of the opposite side and the length of the hypotenuse, you can use the inverse sine function to find the angle.

Q7: What is the relationship between the inverse sine function and the sine function?

A7: The inverse sine function and the sine function are inverse functions of each other. This means that if $\sin \theta = x$, then $\sin^{-1}x = \theta$.

Q8: Can I use the inverse sine function to find the angle of a non-right triangle?

A8: Yes, you can use the inverse sine function to find the angle of a non-right triangle. If you know the length of the opposite side and the length of the adjacent side, you can use the inverse sine function to find the angle.

Conclusion

In conclusion, the inverse sine function is a powerful tool for finding the angle of a given value. It has many real-world applications and can be used to find the angle of a right triangle or a non-right triangle. We hope that this Q&A article has helped you understand the inverse sine function and its applications.

Final Answer

The final answer is: $\boxed{\frac{3 \pi}{2}}$