$\arcsin \left(\frac{1}{2} \sqrt{2}\right) =$

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Introduction

The arcsine function is the inverse of the sine function, and it is used to find the angle whose sine is a given value. In this article, we will explore the value of $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$, which is a specific case of the arcsine function.

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry, and it is used to describe the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. The sine function is denoted by $\sin(x)$, where $x$ is the angle in question.

The Arcsine Function

The arcsine function is the inverse of the sine function, and it is denoted by $\arcsin(x)$. The arcsine function returns the angle whose sine is equal to a given value. In other words, if $\sin(x) = y$, then $\arcsin(y) = x$.

Evaluating $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$

To evaluate $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$, we need to find the angle whose sine is equal to $\frac{1}{2} \sqrt{2}$. We can start by using the Pythagorean identity, which states that $\sin^2(x) + \cos^2(x) = 1$.

Using the Pythagorean Identity

Using the Pythagorean identity, we can rewrite $\sin^2(x) = 1 - \cos^2(x)$. Substituting this into the equation $\sin(x) = \frac{1}{2} \sqrt{2}$, we get:

$$\left(\frac{1}{2} \sqrt{2}\right)^2 = 1 - \cos^2(x)$$

Simplifying this equation, we get:

$$\frac{1}{2} = 1 - \cos^2(x)$$

Solving for $\cos(x)$

To solve for $\cos(x)$, we can rearrange the equation to get:

$$\cos^2(x) = 1 - \frac{1}{2}$$

Simplifying this equation, we get:

$$\cos^2(x) = \frac{1}{2}$$

Taking the square root of both sides, we get:

$$\cos(x) = \pm \frac{1}{\sqrt{2}}$$

Finding the Angle

Since $\cos(x) = \pm \frac{1}{\sqrt{2}}$, we can find the angle $x$ by using the inverse cosine function. The inverse cosine function returns the angle whose cosine is equal to a given value.

Using the Inverse Cosine Function

Using the inverse cosine function, we can find the angle $x$ as follows:

$$x = \cos^{-1}\left(\pm \frac{1}{\sqrt{2}}\right)$$

Evaluating the Inverse Cosine Function

Evaluating the inverse cosine function, we get:

$$x = \frac{\pi}{4} \text{ or } x = \frac{3\pi}{4}$$

Conclusion

In conclusion, the value of $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$ is $\frac{\pi}{4}$ or $\frac{3\pi}{4}$. This is because the sine function is periodic, and the arcsine function returns the principal value of the angle whose sine is equal to a given value.

Final Answer

The final answer is $\boxed{\frac{\pi}{4}}$.

Additional Information

The arcsine function is an important concept in mathematics, and it has many applications in physics, engineering, and other fields. The value of $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$ is a specific case of the arcsine function, and it can be used to solve problems involving right-angled triangles and trigonometric functions.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

• arcsine function
• sine function
• inverse cosine function
• right-angled triangle
• trigonometric functions
• mathematics
• physics
• engineering
  

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Introduction

In our previous article, we explored the value of $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$, which is a specific case of the arcsine function. In this article, we will answer some frequently asked questions about the arcsine function and its applications.

Q&A

Q: What is the arcsine function?

A: The arcsine function is the inverse of the sine function, and it is used to find the angle whose sine is a given value.

Q: How do I evaluate the arcsine function?

A: To evaluate the arcsine function, you need to find the angle whose sine is equal to a given value. You can use the Pythagorean identity to rewrite the equation and solve for the angle.

Q: What is the principal value of the arcsine function?

A: The principal value of the arcsine function is the angle whose sine is equal to a given value, and it lies in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Q: How do I use the arcsine function in real-world applications?

A: The arcsine function has many applications in physics, engineering, and other fields. For example, you can use it to solve problems involving right-angled triangles and trigonometric functions.

Q: What is the relationship between the arcsine function and the inverse cosine function?

A: The arcsine function and the inverse cosine function are related, and they can be used together to solve problems involving right-angled triangles and trigonometric functions.

Q: Can I use the arcsine function to solve problems involving complex numbers?

A: Yes, you can use the arcsine function to solve problems involving complex numbers. However, you need to be careful when working with complex numbers, as they can have multiple values.

Q: How do I evaluate the arcsine function for complex numbers?

A: To evaluate the arcsine function for complex numbers, you need to use the complex logarithm and the complex exponential function. You can also use the inverse tangent function to evaluate the arcsine function for complex numbers.

Q: What is the range of the arcsine function?

A: The range of the arcsine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Q: How do I use the arcsine function to solve problems involving periodic functions?

A: You can use the arcsine function to solve problems involving periodic functions by using the periodicity of the sine function. For example, you can use the arcsine function to find the angle whose sine is equal to a given value, and then use the periodicity of the sine function to find the other angles whose sine is equal to the same value.

Conclusion

In conclusion, the arcsine function is an important concept in mathematics, and it has many applications in physics, engineering, and other fields. By understanding the arcsine function and its applications, you can solve problems involving right-angled triangles and trigonometric functions.

Final Answer

The final answer is $\boxed{\frac{\pi}{4}}$.

Additional Information

The arcsine function is an important concept in mathematics, and it has many applications in physics, engineering, and other fields. The value of $\arcsin \left(\frac{1}{2} \sqrt{2}\right)$ is a specific case of the arcsine function, and it can be used to solve problems involving right-angled triangles and trigonometric functions.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

• arcsine function
• sine function
• inverse cosine function
• right-angled triangle
• trigonometric functions
• mathematics
• physics
• engineering