Find The Exact Value Of \sin^{-1}\left(\frac{\sqrt{2}}{2}\right ]. Sin ⁡ − 1 ( 2 2 ) = □ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \square Sin − 1 ( 2 2 ​ ​ ) = □ (Type Your Answer In Radians.)

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Introduction

In mathematics, the inverse sine function, denoted by $\sin^{-1}$, is a function that returns the angle whose sine is a given value. In this article, we will focus on finding the exact value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$, which is a fundamental problem in trigonometry.

Understanding the Inverse Sine Function

The inverse sine function is defined as the angle whose sine is equal to a given value. In other words, if $\sin(x) = y$, then $\sin^{-1}(y) = x$. The range of the inverse sine function is restricted to the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ to ensure that the function is one-to-one.

The Problem at Hand

We are given the expression $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ and we need to find its exact value. To do this, we can use the definition of the inverse sine function and the properties of the sine function.

Using the Definition of the Inverse Sine Function

By definition, $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ is the angle whose sine is equal to $\frac{\sqrt{2}}{2}$. We know that the sine of $\frac{\pi}{4}$ is equal to $\frac{\sqrt{2}}{2}$. Therefore, we can conclude that $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$.

Verifying the Answer

To verify our answer, we can use the fact that the sine function is periodic with period $2\pi$. This means that if $\sin(x) = y$, then $\sin(x + 2\pi k) = y$ for any integer $k$. In our case, we have $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$, and we can verify that $\sin\left(\frac{\pi}{4} + 2\pi k\right) = \frac{\sqrt{2}}{2}$ for any integer $k$.

Conclusion

In conclusion, we have found the exact value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ to be $\frac{\pi}{4}$. This result is consistent with the definition of the inverse sine function and the properties of the sine function.

Additional Examples

Here are a few additional examples of finding the exact value of the inverse sine function:

• $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$
• $\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$
• $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

Applications of the Inverse Sine Function

The inverse sine function has many applications in mathematics and science. For example, it is used to find the angle of elevation of a building or the angle of depression of a projectile. It is also used in the calculation of trigonometric identities and in the solution of trigonometric equations.

Final Thoughts

In conclusion, finding the exact value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ is a fundamental problem in trigonometry. By using the definition of the inverse sine function and the properties of the sine function, we have found the exact value to be $\frac{\pi}{4}$. This result has many applications in mathematics and science, and it is an important tool for solving problems in trigonometry.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Identities" by Paul Dawkins

Further Reading

• [1] "Inverse Trigonometric Functions" by Paul Dawkins
• [2] "Trigonometry" by I. M. Gelfand
• [3] "Calculus" by James Stewart
  

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Introduction

In our previous article, we discussed how to find the exact value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$. In this article, we will answer some frequently asked questions related to this topic.

Q1: What is the inverse sine function?

A1: 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 $\sin(x) = y$, then $\sin^{-1}(y) = x$.

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. This means that for each value of $y$, there is only one value of $x$ such that $\sin(x) = y$.

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

A3: We can find the exact value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ by using the definition of the inverse sine function and the properties of the sine function. Specifically, we know that the sine of $\frac{\pi}{4}$ is equal to $\frac{\sqrt{2}}{2}$, so we can conclude that $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$.

Q4: What are some common values of the inverse sine function?

A4: Some common values of the inverse sine function include:

• $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$
• $\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$
• $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

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

A5: The inverse sine function has many applications in mathematics and science. For example, it is used to find the angle of elevation of a building or the angle of depression of a projectile. It is also used in the calculation of trigonometric identities and in the solution of trigonometric equations.

Q6: What are some common mistakes to avoid when working with the inverse sine function?

A6: Some common mistakes to avoid when working with the inverse sine function include:

• Not checking the range of the function
• Not using the correct definition of the inverse sine function
• Not simplifying the expression before evaluating it

Q7: How do we evaluate the inverse sine function using a calculator?

A7: To evaluate the inverse sine function using a calculator, you can use the $\sin^{-1}$ button on the calculator. For example, to find the value of $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$, you would enter $\sin^{-1}(\frac{\sqrt{2}}{2})$ and press the $\sin^{-1}$ button.

Q8: What are some advanced topics related to the inverse sine function?

A8: Some advanced topics related to the inverse sine function include:

• The inverse cosine function
• The inverse tangent function
• The hyperbolic functions

Conclusion

In conclusion, the inverse sine function is a fundamental concept in trigonometry that has many applications in mathematics and science. By understanding the definition and properties of the inverse sine function, we can solve problems and evaluate expressions involving this function.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Identities" by Paul Dawkins

Further Reading

• [1] "Inverse Trigonometric Functions" by Paul Dawkins
• [2] "Trigonometry" by I. M. Gelfand
• [3] "Calculus" by James Stewart