What Is The Value Of \arccos \left(-\frac{1}{2}\right ] In Radians?A. Π 6 \frac{\pi}{6} 6 Π ​ B. 5 Π 6 \frac{5\pi}{6} 6 5 Π ​ C. Π 3 \frac{\pi}{3} 3 Π ​ D. 2 Π 3 \frac{2\pi}{3} 3 2 Π ​

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Introduction

In mathematics, the arccosine function, denoted as $\arccos x$, is the inverse of the cosine function. It returns the angle whose cosine is a given number. In this article, we will explore the value of $\arccos \left(-\frac{1}{2}\right)$ in radians.

Understanding the Arccosine Function

The arccosine function is defined as the inverse of the cosine function. This means that if $\cos \theta = x$, then $\arccos x = \theta$. The arccosine function is a multivalued function, meaning that it has multiple values for a given input. However, in most mathematical contexts, we restrict the range of the arccosine function to the interval $[0, \pi]$.

The Value of $\arccos \left(-\frac{1}{2}\right)$

To find the value of $\arccos \left(-\frac{1}{2}\right)$, we need to find the angle whose cosine is $-\frac{1}{2}$. We can use the unit circle to visualize this.

Using the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The cosine of an angle is equal to the x-coordinate of the point on the unit circle corresponding to that angle. Since the cosine of $\arccos \left(-\frac{1}{2}\right)$ is $-\frac{1}{2}$, we need to find the point on the unit circle with an x-coordinate of $-\frac{1}{2}$.

Finding the Angle

We can find the angle by using the inverse cosine function. However, we need to be careful because the inverse cosine function is not defined for all values of x. In this case, we are looking for the angle whose cosine is $-\frac{1}{2}$.

Using Trigonometric Identities

We can use trigonometric identities to find the value of $\arccos \left(-\frac{1}{2}\right)$. One such identity is the cosine of the sum of two angles:

$$\cos (A + B) = \cos A \cos B - \sin A \sin B$$

We can use this identity to find the value of $\arccos \left(-\frac{1}{2}\right)$.

Finding the Value

Using the identity above, we can write:

$$\cos \left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \cos \frac{\pi}{3} \cos \frac{2\pi}{3} - \sin \frac{\pi}{3} \sin \frac{2\pi}{3}$$

Simplifying this expression, we get:

$$\cos \left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \left(-\frac{1}{2}\right)$$

This means that the angle whose cosine is $-\frac{1}{2}$ is $\frac{\pi}{3} + \frac{2\pi}{3} = \frac{5\pi}{6}$.

Conclusion

In conclusion, the value of $\arccos \left(-\frac{1}{2}\right)$ in radians is $\frac{5\pi}{6}$. This can be verified using the unit circle and trigonometric identities.

Final Answer

The final answer is $\boxed{\frac{5\pi}{6}}$.

Discussion

The discussion of this problem involves understanding the arccosine function, the unit circle, and trigonometric identities. It requires the ability to visualize and manipulate mathematical concepts to arrive at the solution.

Related Problems

Some related problems that involve the arccosine function include:

• Finding the value of $\arccos \left(\frac{1}{2}\right)$ in radians
• Finding the value of $\arccos \left(-\frac{\sqrt{3}}{2}\right)$ in radians
• Finding the value of $\arccos \left(\frac{\sqrt{3}}{2}\right)$ in radians

These problems require the same level of understanding and manipulation of mathematical concepts as the original problem.

References

• [1] "Arccosine Function" by MathWorld
• [2] "Unit Circle" by MathWorld
• [3] "Trigonometric Identities" by MathWorld

Note: The references provided are for general information and are not specific to the problem at hand. They are included to provide additional context and resources for the reader.

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Introduction

In our previous article, we explored the value of $\arccos \left(-\frac{1}{2}\right)$ in radians. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the arccosine function?

A: The arccosine function, denoted as $\arccos x$, is the inverse of the cosine function. It returns the angle whose cosine is a given number.

Q: Why is the arccosine function important?

A: The arccosine function is important in mathematics because it is used to find the angle whose cosine is a given number. This is useful in a variety of applications, including trigonometry, calculus, and engineering.

Q: How do I find the value of $\arccos \left(-\frac{1}{2}\right)$ in radians?

A: To find the value of $\arccos \left(-\frac{1}{2}\right)$ in radians, you can use the unit circle and trigonometric identities. Specifically, you can use the identity $\cos \left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \left(-\frac{1}{2}\right)$ to find the value.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The cosine of an angle is equal to the x-coordinate of the point on the unit circle corresponding to that angle.

Q: How do I use the unit circle to find the value of $\arccos \left(-\frac{1}{2}\right)$ in radians?

A: To use the unit circle to find the value of $\arccos \left(-\frac{1}{2}\right)$ in radians, you can visualize the point on the unit circle with an x-coordinate of $-\frac{1}{2}$. This point corresponds to the angle $\frac{5\pi}{6}$.

Q: What are some common mistakes to avoid when finding the value of $\arccos \left(-\frac{1}{2}\right)$ in radians?

A: Some common mistakes to avoid when finding the value of $\arccos \left(-\frac{1}{2}\right)$ in radians include:

• Not using the correct trigonometric identity
• Not visualizing the point on the unit circle correctly
• Not considering the range of the arccosine function

Q: How do I verify the value of $\arccos \left(-\frac{1}{2}\right)$ in radians?

A: To verify the value of $\arccos \left(-\frac{1}{2}\right)$ in radians, you can use the unit circle and trigonometric identities. Specifically, you can use the identity $\cos \left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \left(-\frac{1}{2}\right)$ to verify the value.

Q: What are some related problems to the value of $\arccos \left(-\frac{1}{2}\right)$ in radians?

A: Some related problems to the value of $\arccos \left(-\frac{1}{2}\right)$ in radians include:

• Finding the value of $\arccos \left(\frac{1}{2}\right)$ in radians
• Finding the value of $\arccos \left(-\frac{\sqrt{3}}{2}\right)$ in radians
• Finding the value of $\arccos \left(\frac{\sqrt{3}}{2}\right)$ in radians

Conclusion

In conclusion, the value of $\arccos \left(-\frac{1}{2}\right)$ in radians is $\frac{5\pi}{6}$. This can be verified using the unit circle and trigonometric identities. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions related to this topic.

Final Answer

The final answer is $\boxed{\frac{5\pi}{6}}$.

Discussion

The discussion of this problem involves understanding the arccosine function, the unit circle, and trigonometric identities. It requires the ability to visualize and manipulate mathematical concepts to arrive at the solution.

Related Problems

Some related problems that involve the arccosine function include:

• Finding the value of $\arccos \left(\frac{1}{2}\right)$ in radians
• Finding the value of $\arccos \left(-\frac{\sqrt{3}}{2}\right)$ in radians
• Finding the value of $\arccos \left(\frac{\sqrt{3}}{2}\right)$ in radians

These problems require the same level of understanding and manipulation of mathematical concepts as the original problem.

References

• [1] "Arccosine Function" by MathWorld
• [2] "Unit Circle" by MathWorld
• [3] "Trigonometric Identities" by MathWorld

Note: The references provided are for general information and are not specific to the problem at hand. They are included to provide additional context and resources for the reader.