Find The Exact Value Of The Expression:$\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right\]

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Introduction

In this article, we will delve into the world of trigonometry and explore the concept of inverse trigonometric functions. We will examine the given expression, $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$, and find its exact value. This involves understanding the properties of inverse trigonometric functions, specifically the tangent and cosine inverse functions.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle whose trigonometric function is a given value. For example, the tangent inverse function, denoted as $\tan^{-1}x$, gives the angle whose tangent is $x$. Similarly, the cosine inverse function, denoted as $\cos^{-1}x$, gives the angle whose cosine is $x$.

The Tangent Inverse Function

The tangent inverse function, $\tan^{-1}x$, is defined as the angle $\theta$ such that $\tan\theta = x$. This function is also known as the arctangent function. The range of the tangent inverse function is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

The Cosine Inverse Function

The cosine inverse function, $\cos^{-1}x$, is defined as the angle $\theta$ such that $\cos\theta = x$. This function is also known as the arccosine function. The range of the cosine inverse function is $[0, \pi]$.

The Given Expression

The given expression is $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$. To find the exact value of this expression, we need to evaluate the expression inside the brackets first.

Evaluating the Expression Inside the Brackets

Let's start by evaluating the expression $\tan^{-1} \frac{24}{7}$. We can do this by drawing a right triangle with legs 24 and 7, and hypotenuse $h$. Using the Pythagorean theorem, we can find the value of $h$.

Calculating the Hypotenuse

Using the Pythagorean theorem, we have:

$$h^2 = 24^2 + 7^2$$

$$h^2 = 576 + 49$$

$$h^2 = 625$$

$$h = \sqrt{625}$$

$$h = 25$$

Finding the Angle

Now that we have the value of the hypotenuse, we can find the angle $\theta$ using the tangent function:

$$\tan\theta = \frac{24}{7}$$

$$\theta = \tan^{-1} \frac{24}{7}$$

Evaluating the Cosine Inverse Function

Next, let's evaluate the expression $\cos^{-1} \frac{3}{5}$. We can do this by drawing a right triangle with legs 3 and 4, and hypotenuse 5. Using the Pythagorean theorem, we can verify that this is a valid right triangle.

Verifying the Right Triangle

Using the Pythagorean theorem, we have:

$$5^2 = 3^2 + 4^2$$

$$25 = 9 + 16$$

$$25 = 25$$

This confirms that the triangle is a valid right triangle.

Finding the Angle

Now that we have the value of the hypotenuse, we can find the angle $\phi$ using the cosine function:

$$\cos\phi = \frac{3}{5}$$

$$\phi = \cos^{-1} \frac{3}{5}$$

Evaluating the Expression Inside the Brackets

Now that we have evaluated the expressions $\tan^{-1} \frac{24}{7}$ and $\cos^{-1} \frac{3}{5}$, we can substitute these values into the original expression:

$$\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$$

$$\cos \left[\theta - \phi\right]$$

Using the Cosine Angle Addition Formula

We can use the cosine angle addition formula to simplify the expression:

$$\cos \left[\theta - \phi\right] = \cos\theta\cos\phi + \sin\theta\sin\phi$$

Substituting the Values

Now that we have the values of $\theta$ and $\phi$, we can substitute them into the expression:

$$\cos\theta\cos\phi + \sin\theta\sin\phi$$

$$\left(\frac{24}{25}\right)\left(\frac{3}{5}\right) + \left(\frac{7}{25}\right)\left(\frac{4}{5}\right)$$

Simplifying the Expression

Now that we have the expression simplified, we can evaluate it:

$$\left(\frac{24}{25}\right)\left(\frac{3}{5}\right) + \left(\frac{7}{25}\right)\left(\frac{4}{5}\right)$$

$$\frac{72}{125} + \frac{28}{125}$$

$$\frac{100}{125}$$

$$\frac{4}{5}$$

Conclusion

In this article, we have found the exact value of the expression $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$. We have used the properties of inverse trigonometric functions, specifically the tangent and cosine inverse functions, to evaluate the expression. The final answer is $\boxed{\frac{4}{5}}$.

Final Answer

The final answer is $\boxed{\frac{4}{5}}$.

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Introduction

In our previous article, we explored the concept of inverse trigonometric functions and used them to find the exact value of the expression $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the tangent inverse function?

A: The tangent inverse function, denoted as $\tan^{-1}x$, is defined as the angle $\theta$ such that $\tan\theta = x$. This function is also known as the arctangent function.

Q: What is the range of the tangent inverse function?

A: The range of the tangent inverse function is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Q: What is the cosine inverse function?

A: The cosine inverse function, denoted as $\cos^{-1}x$, is defined as the angle $\theta$ such that $\cos\theta = x$. This function is also known as the arccosine function.

Q: What is the range of the cosine inverse function?

A: The range of the cosine inverse function is $[0, \pi]$.

Q: How do I evaluate the expression $\tan^{-1} \frac{24}{7}$?

A: To evaluate the expression $\tan^{-1} \frac{24}{7}$, you can draw a right triangle with legs 24 and 7, and hypotenuse $h$. Using the Pythagorean theorem, you can find the value of $h$.

Q: How do I find the angle $\theta$ using the tangent function?

A: To find the angle $\theta$ using the tangent function, you can use the formula $\tan\theta = \frac{24}{7}$.

Q: How do I evaluate the expression $\cos^{-1} \frac{3}{5}$?

A: To evaluate the expression $\cos^{-1} \frac{3}{5}$, you can draw a right triangle with legs 3 and 4, and hypotenuse 5. Using the Pythagorean theorem, you can verify that this is a valid right triangle.

Q: How do I find the angle $\phi$ using the cosine function?

A: To find the angle $\phi$ using the cosine function, you can use the formula $\cos\phi = \frac{3}{5}$.

Q: How do I use the cosine angle addition formula to simplify the expression?

A: To use the cosine angle addition formula to simplify the expression, you can use the formula $\cos \left[\theta - \phi\right] = \cos\theta\cos\phi + \sin\theta\sin\phi$.

Q: What is the final answer to the expression $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$?

A: The final answer to the expression $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$ is $\boxed{\frac{4}{5}}$.

Conclusion

In this article, we have answered some of the most frequently asked questions related to finding the exact value of the expression $\cos \left[\tan ^{-1} \frac{24}{7} - \cos ^{-1} \frac{3}{5}\right]$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.

Final Answer

The final answer is $\boxed{\frac{4}{5}}$.