Evaluate The Expression:$\arccos \left(\frac{1}{2} \sqrt{3}\right$\] = ?

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Introduction

In mathematics, the arccosine function is the inverse of the cosine function. It returns the angle whose cosine is a given number. The arccosine function is denoted by $\arccos x$ and is defined as the angle $\theta$ in the range $[0, \pi]$ such that $\cos \theta = x$. In this article, we will evaluate the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$.

Understanding the Arccosine Function

The arccosine function is a trigonometric function that is used to find the angle whose cosine is a given number. It is defined as the angle $\theta$ in the range $[0, \pi]$ such that $\cos \theta = x$. The arccosine function is the inverse of the cosine function, which means that if $\cos \theta = x$, then $\arccos x = \theta$.

Evaluating the Expression

To evaluate the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$, we need to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$. We can use the unit circle to find this angle.

Using the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions, including the cosine function. We can use the unit circle to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$.

Finding the Angle

To find the angle whose cosine is $\frac{1}{2} \sqrt{3}$, we need to find the point on the unit circle that has a cosine of $\frac{1}{2} \sqrt{3}$. We can do this by using the coordinates of the point on the unit circle.

Using the Coordinates of the Point

The coordinates of the point on the unit circle are given by $(\cos \theta, \sin \theta)$. We can use these coordinates to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$.

Finding the Angle Whose Cosine is $\frac{1}{2} \sqrt{3}$

To find the angle whose cosine is $\frac{1}{2} \sqrt{3}$, we need to find the value of $\theta$ such that $\cos \theta = \frac{1}{2} \sqrt{3}$. We can do this by using the coordinates of the point on the unit circle.

Using the Pythagorean Identity

The Pythagorean identity is a fundamental identity in trigonometry that relates the sine and cosine functions. It states that $\sin^2 \theta + \cos^2 \theta = 1$. We can use this identity to find the value of $\theta$ such that $\cos \theta = \frac{1}{2} \sqrt{3}$.

Finding the Value of $\theta$

To find the value of $\theta$ such that $\cos \theta = \frac{1}{2} \sqrt{3}$, we need to use the Pythagorean identity. We can do this by substituting $\cos \theta = \frac{1}{2} \sqrt{3}$ into the Pythagorean identity.

Solving for $\theta$

To solve for $\theta$, we need to isolate $\theta$ in the equation. We can do this by using algebraic manipulations.

Finding the Angle Whose Cosine is $\frac{1}{2} \sqrt{3}$

After solving for $\theta$, we find that the angle whose cosine is $\frac{1}{2} \sqrt{3}$ is $\frac{\pi}{6}$.

Conclusion

In this article, we evaluated the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$. We used the unit circle and the Pythagorean identity to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$. We found that the angle whose cosine is $\frac{1}{2} \sqrt{3}$ is $\frac{\pi}{6}$.

Final Answer

The final answer is $\boxed{\frac{\pi}{6}}$.

Related Topics

• Arccosine function
• Unit circle
• Pythagorean identity
• Trigonometric functions

References

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

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

In our previous article, we evaluated the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ and found that the angle whose cosine is $\frac{1}{2} \sqrt{3}$ is $\frac{\pi}{6}$. In this article, we will answer some frequently asked questions about evaluating the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$.

Q1: What is the arccosine function?

A1: The arccosine function is the inverse of the cosine function. It returns the angle whose cosine is a given number. The arccosine function is denoted by $\arccos x$ and is defined as the angle $\theta$ in the range $[0, \pi]$ such that $\cos \theta = x$.

Q2: How do I evaluate the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$?

A2: To evaluate the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$, you need to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$. You can use the unit circle and the Pythagorean identity to find this angle.

Q3: What is the unit circle?

A3: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions, including the cosine function.

Q4: How do I use the unit circle to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$?

A4: To use the unit circle to find the angle whose cosine is $\frac{1}{2} \sqrt{3}$, you need to find the point on the unit circle that has a cosine of $\frac{1}{2} \sqrt{3}$. You can do this by using the coordinates of the point on the unit circle.

Q5: What is the Pythagorean identity?

A5: The Pythagorean identity is a fundamental identity in trigonometry that relates the sine and cosine functions. It states that $\sin^2 \theta + \cos^2 \theta = 1$. You can use this identity to find the value of $\theta$ such that $\cos \theta = \frac{1}{2} \sqrt{3}$.

Q6: How do I use the Pythagorean identity to find the value of $\theta$?

A6: To use the Pythagorean identity to find the value of $\theta$, you need to substitute $\cos \theta = \frac{1}{2} \sqrt{3}$ into the Pythagorean identity. You can then solve for $\theta$ using algebraic manipulations.

Q7: What is the final answer to the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$?

A7: The final answer to the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ is $\boxed{\frac{\pi}{6}}$.

Q8: What are some related topics to the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$?

A8: Some related topics to the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ include the arccosine function, the unit circle, the Pythagorean identity, and trigonometric functions.

Q9: Where can I find more information about the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$?

A9: You can find more information about the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ in textbooks on trigonometry and calculus, as well as online resources such as Khan Academy and Wolfram Alpha.

Q10: How can I apply the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ in real-world problems?

A10: You can apply the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$ in real-world problems such as finding the angle of elevation of a building, the angle of depression of a well, or the angle of a triangle.

Conclusion

In this article, we answered some frequently asked questions about evaluating the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$. We hope that this article has been helpful in understanding the concept of the arccosine function and how to evaluate the expression $\arccos \left(\frac{1}{2} \sqrt{3}\right)$.