Simplify The Expression: $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right$\]

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Introduction

In mathematics, trigonometric functions and their inverses play a crucial role in solving various problems. The given expression involves the inverse cosine function, which is denoted by $\cos^{-1}$. The expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$ seems complex, but it can be simplified using the properties of trigonometric functions. In this article, we will explore the simplification of this expression and provide a step-by-step solution.

Understanding the Inverse Cosine Function

The inverse cosine function, denoted by $\cos^{-1}$, is a function that returns the angle whose cosine is a given value. In other words, if $\cos(x) = y$, then $\cos^{-1}(y) = x$. The range of the inverse cosine function is $[0, \pi]$.

Simplifying the Expression

To simplify the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$, we need to evaluate the innermost function first, which is $\cos\left(-\frac{13 \pi}{8}\right)$. We can use the periodicity of the cosine function to simplify this expression.

Periodicity of the Cosine Function

The cosine function has a period of $2\pi$, which means that $\cos(x) = \cos(x + 2k\pi)$, where $k$ is an integer. We can use this property to simplify the expression $\cos\left(-\frac{13 \pi}{8}\right)$.

Evaluating the Innermost Function

Using the periodicity of the cosine function, we can rewrite $\cos\left(-\frac{13 \pi}{8}\right)$ as $\cos\left(-\frac{13 \pi}{8} + 2k\pi\right)$. We want to find a value of $k$ such that the argument of the cosine function is in the range $[0, 2\pi]$.

Finding the Appropriate Value of $k$

To find the appropriate value of $k$, we need to add or subtract multiples of $2\pi$ from the argument of the cosine function until it is in the range $[0, 2\pi]$. Let's try adding $2\pi$ to the argument:

$$-\frac{13 \pi}{8} + 2k\pi = -\frac{13 \pi}{8} + 2\pi$$

Simplifying the Argument

We can simplify the argument by combining the fractions:

$$-\frac{13 \pi}{8} + 2\pi = -\frac{13 \pi}{8} + \frac{16 \pi}{8} = \frac{3 \pi}{8}$$

Evaluating the Cosine Function

Now that we have simplified the argument, we can evaluate the cosine function:

$$\cos\left(\frac{3 \pi}{8}\right) = \cos\left(2\pi - \frac{11 \pi}{8}\right)$$

Using the Periodicity of the Cosine Function Again

We can use the periodicity of the cosine function again to simplify the expression:

$$\cos\left(2\pi - \frac{11 \pi}{8}\right) = \cos\left(-\frac{11 \pi}{8}\right)$$

Evaluating the Cosine Function Again

Now that we have simplified the argument, we can evaluate the cosine function:

$$\cos\left(-\frac{11 \pi}{8}\right) = \cos\left(\frac{11 \pi}{8}\right)$$

Using the Symmetry of the Cosine Function

The cosine function is symmetric about the origin, which means that $\cos(x) = \cos(-x)$. We can use this property to simplify the expression:

$$\cos\left(\frac{11 \pi}{8}\right) = \cos\left(2\pi - \frac{3 \pi}{8}\right)$$

Evaluating the Cosine Function Again

Now that we have simplified the argument, we can evaluate the cosine function:

$$\cos\left(2\pi - \frac{3 \pi}{8}\right) = \cos\left(-\frac{3 \pi}{8}\right)$$

Using the Periodicity of the Cosine Function Again

We can use the periodicity of the cosine function again to simplify the expression:

$$\cos\left(-\frac{3 \pi}{8}\right) = \cos\left(\frac{3 \pi}{8}\right)$$

Evaluating the Cosine Function Again

Now that we have simplified the argument, we can evaluate the cosine function:

$$\cos\left(\frac{3 \pi}{8}\right) = \frac{1}{2}$$

Simplifying the Expression

Now that we have evaluated the innermost function, we can simplify the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$:

$$\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right) = \cos^{-1}\left(\frac{1}{2}\right)$$

Evaluating the Inverse Cosine Function

The inverse cosine function returns the angle whose cosine is a given value. In this case, the given value is $\frac{1}{2}$, which corresponds to an angle of $\frac{\pi}{3}$.

Conclusion

In conclusion, the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$ can be simplified using the properties of trigonometric functions. By evaluating the innermost function and using the periodicity and symmetry of the cosine function, we can simplify the expression to $\cos^{-1}\left(\frac{1}{2}\right)$, which corresponds to an angle of $\frac{\pi}{3}$.

Final Answer

The final answer is $\boxed{\frac{\pi}{3}}$.

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Introduction

In our previous article, we explored the simplification of the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$. We used the properties of trigonometric functions, including the periodicity and symmetry of the cosine function, to simplify the expression. In this article, we will answer some common questions related to this topic.

Q: What is the inverse cosine function?

A: The inverse cosine function, denoted by $\cos^{-1}$, is a function that returns the angle whose cosine is a given value. In other words, if $\cos(x) = y$, then $\cos^{-1}(y) = x$. The range of the inverse cosine function is $[0, \pi]$.

Q: How do you simplify the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$?

A: To simplify the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$, we need to evaluate the innermost function first, which is $\cos\left(-\frac{13 \pi}{8}\right)$. We can use the periodicity of the cosine function to simplify this expression.

Q: What is the periodicity of the cosine function?

A: The cosine function has a period of $2\pi$, which means that $\cos(x) = \cos(x + 2k\pi)$, where $k$ is an integer. We can use this property to simplify the expression $\cos\left(-\frac{13 \pi}{8}\right)$.

Q: How do you find the appropriate value of $k$ to simplify the expression $\cos\left(-\frac{13 \pi}{8}\right)$?

A: To find the appropriate value of $k$, we need to add or subtract multiples of $2\pi$ from the argument of the cosine function until it is in the range $[0, 2\pi]$. Let's try adding $2\pi$ to the argument:

$$-\frac{13 \pi}{8} + 2k\pi = -\frac{13 \pi}{8} + 2\pi$$

Q: How do you simplify the argument $-\frac{13 \pi}{8} + 2\pi$?

A: We can simplify the argument by combining the fractions:

$$-\frac{13 \pi}{8} + 2\pi = -\frac{13 \pi}{8} + \frac{16 \pi}{8} = \frac{3 \pi}{8}$$

Q: What is the final answer to the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$?

A: The final answer to the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$ is $\boxed{\frac{\pi}{3}}$.

Q: What are some common mistakes to avoid when simplifying the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$?

A: Some common mistakes to avoid when simplifying the expression $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$ include:

• Not using the periodicity of the cosine function to simplify the expression
• Not finding the appropriate value of $k$ to simplify the expression
• Not simplifying the argument correctly
• Not using the symmetry of the cosine function to simplify the expression

Q: How can I practice simplifying expressions like $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$?

A: You can practice simplifying expressions like $\cos^{-1}\left(\cos\left(-\frac{13 \pi}{8}\right)\right)$ by working through examples and exercises in a textbook or online resource. You can also try simplifying different expressions on your own and checking your answers with a calculator or online tool.

Q: What are some real-world applications of the inverse cosine function?

A: The inverse cosine function has many real-world applications, including:

• Calculating the angle of elevation of a building or a mountain
• Determining the angle of a triangle given the lengths of two sides
• Calculating the cosine of an angle given the sine or tangent of the angle
• Solving problems in physics, engineering, and computer science that involve trigonometry and inverse trigonometric functions.