Find The Exact Value Of \cos^{-1}\left(\cos \frac{\pi}{6}\right ].Write Your Answer In Radians In Terms Of Π \pi Π .

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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 $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$. Inverse trigonometric functions are used to find the angle whose trigonometric function is a given value. In this case, we are given the value of $\cos \frac{\pi}{6}$ and we need to find the angle whose cosine is this value.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are denoted by the prefix "arc" or "inverse". For example, the inverse of the cosine function is denoted by $\cos^{-1}$. The inverse trigonometric functions are used to find the angle whose trigonometric function is a given value.

Properties of Inverse Trigonometric Functions

Inverse trigonometric functions have several properties that are important to understand. Some of these properties are:

• The range of an inverse trigonometric function is restricted to a specific interval.
• The domain of an inverse trigonometric function is restricted to a specific interval.
• The inverse trigonometric functions are not one-to-one functions, meaning that they do not pass the horizontal line test.

Finding the Exact Value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$

To find the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$, we need to understand the properties of the inverse cosine function. The inverse cosine function is defined as the angle whose cosine is a given value.

Using the Definition of the Inverse Cosine Function

The inverse cosine function is defined as:

$$\cos^{-1}x = \theta \iff \cos \theta = x$$

where $\theta$ is the angle whose cosine is $x$.

Applying the Definition to the Given Problem

In this case, we are given the value of $\cos \frac{\pi}{6}$ and we need to find the angle whose cosine is this value. Using the definition of the inverse cosine function, we can write:

$$\cos^{-1}\left(\cos \frac{\pi}{6}\right) = \theta \iff \cos \theta = \cos \frac{\pi}{6}$$

Solving for $\theta$

Since the cosine function is periodic, we can add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{6}$ without changing the value of the cosine function. Therefore, we can write:

$$\cos \theta = \cos \left(\frac{\pi}{6} + 2k\pi\right)$$

where $k$ is an integer.

Finding the Value of $\theta$

To find the value of $\theta$, we need to find the angle whose cosine is $\cos \frac{\pi}{6}$. Since the cosine function is positive in the first and fourth quadrants, we can write:

$$\cos \theta = \cos \left(\frac{\pi}{6} + 2k\pi\right) = \cos \left(\frac{\pi}{6}\right)$$

This implies that:

$$\theta = \frac{\pi}{6} + 2k\pi$$

Finding the Value of $k$

To find the value of $k$, we need to find the smallest positive integer that satisfies the equation:

$$\frac{\pi}{6} + 2k\pi = \theta$$

Since the cosine function is periodic, we can add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{6}$ without changing the value of the cosine function. Therefore, we can write:

$$\frac{\pi}{6} + 2k\pi = \theta \iff \frac{\pi}{6} = \theta - 2k\pi$$

Solving for $k$

To solve for $k$, we need to find the smallest positive integer that satisfies the equation:

$$\frac{\pi}{6} = \theta - 2k\pi$$

Since the cosine function is positive in the first and fourth quadrants, we can write:

$$\frac{\pi}{6} = \theta - 2k\pi \iff \frac{\pi}{6} = \left(\frac{\pi}{6} + 2k\pi\right) - 2k\pi$$

This implies that:

$$\frac{\pi}{6} = 2k\pi - 2k\pi$$

This equation is satisfied when $k = 0$.

Finding the Value of $\theta$

Now that we have found the value of $k$, we can find the value of $\theta$ by substituting $k = 0$ into the equation:

$$\theta = \frac{\pi}{6} + 2k\pi$$

This gives us:

$$\theta = \frac{\pi}{6} + 2(0)\pi = \frac{\pi}{6}$$

Conclusion

In this article, we have found the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$ by using the definition of the inverse cosine function and the properties of the cosine function. We have shown that:

$$\cos^{-1}\left(\cos \frac{\pi}{6}\right) = \frac{\pi}{6}$$

This result is consistent with the fact that the inverse cosine function is the angle whose cosine is a given value.

Final Answer

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

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Introduction

In our previous article, we explored the concept of inverse trigonometric functions and how to find the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$. In this article, we will answer some common questions related to this topic.

Q1: What is the definition of the inverse cosine function?

A1: The inverse cosine function is defined as the angle whose cosine is a given value. It is denoted by $\cos^{-1}x$ and is defined as:

$$\cos^{-1}x = \theta \iff \cos \theta = x$$

Q2: How do I find the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$?

A2: To find the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$, you need to use the definition of the inverse cosine function and the properties of the cosine function. You can start by writing:

$$\cos^{-1}\left(\cos \frac{\pi}{6}\right) = \theta \iff \cos \theta = \cos \frac{\pi}{6}$$

Then, you can use the fact that the cosine function is periodic to add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{6}$ without changing the value of the cosine function.

Q3: Why do we need to add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{6}$?

A3: We need to add or subtract multiples of $2\pi$ to the angle $\frac{\pi}{6}$ because the cosine function is periodic. This means that the value of the cosine function repeats every $2\pi$ radians. By adding or subtracting multiples of $2\pi$ to the angle $\frac{\pi}{6}$, we can find an angle whose cosine is equal to $\cos \frac{\pi}{6}$.

Q4: How do I find the value of $k$ in the equation $\theta = \frac{\pi}{6} + 2k\pi$?

A4: To find the value of $k$, you need to find the smallest positive integer that satisfies the equation:

$$\frac{\pi}{6} = \theta - 2k\pi$$

You can do this by using the fact that the cosine function is positive in the first and fourth quadrants. This means that the angle $\theta$ must be in one of these quadrants.

Q5: What is the final answer to the problem?

A5: The final answer to the problem is $\boxed{\frac{\pi}{6}}$. This is the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$.

Q6: Can I use a calculator to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$?

A6: Yes, you can use a calculator to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$. However, keep in mind that the calculator may give you an approximate value instead of the exact value.

Q7: How do I know if the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$ is exact or approximate?

A7: You can tell if the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$ is exact or approximate by looking at the calculator's display. If the value is displayed as a decimal approximation, it is likely an approximate value. If the value is displayed as a fraction or a multiple of $\pi$, it is likely an exact value.

Q8: Can I use the inverse cosine function to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$?

A8: Yes, you can use the inverse cosine function to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$. The inverse cosine function is defined as:

$$\cos^{-1}x = \theta \iff \cos \theta = x$$

You can use this definition to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$.

Q9: How do I use the inverse cosine function to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$?

A9: To use the inverse cosine function to find the value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$, you need to follow these steps:

1. Write the equation $\cos \theta = \cos \frac{\pi}{6}$.
2. Use the definition of the inverse cosine function to rewrite the equation as $\cos^{-1}\left(\cos \frac{\pi}{6}\right) = \theta$.
3. Solve for $\theta$ by using the properties of the cosine function.

Q10: What is the final answer to the problem?

A10: The final answer to the problem is $\boxed{\frac{\pi}{6}}$. This is the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$.

Conclusion

In this article, we have answered some common questions related to finding the exact value of $\cos^{-1}\left(\cos \frac{\pi}{6}\right)$. We have shown that the final answer to the problem is $\boxed{\frac{\pi}{6}}$.