Use A Sketch To Find The Exact Value Of The Following Expression:$\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right\]

Introduction

In this article, we will explore how to use a sketch to find the exact value of the expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$. This involves understanding the properties of trigonometric functions, particularly the inverse tangent function, and how to visualize the problem using a sketch.

Understanding the Inverse Tangent Function

The inverse tangent function, denoted as $\tan^{-1}x$, is the angle whose tangent is equal to $x$. In other words, if $\tan \theta = x$, then $\theta = \tan^{-1}x$. The range of the inverse tangent function is typically taken to be $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Sketching the Problem

To sketch the problem, we need to consider the graph of the tangent function and the line $y = -\frac{2}{7}$. The point of intersection between the graph of the tangent function and the line $y = -\frac{2}{7}$ will give us the value of $\tan^{-1}\left(-\frac{2}{7}\right)$.

Finding the Value of $\tan^{-1}\left(-\frac{2}{7}\right)$

Using a graphing calculator or a sketch, we can find the point of intersection between the graph of the tangent function and the line $y = -\frac{2}{7}$. Let's call the angle corresponding to this point of intersection $\theta$. Then, we have $\tan \theta = -\frac{2}{7}$.

Using the Pythagorean Identity

We can use the Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$ to find the value of $\cos \theta$. Since $\tan \theta = -\frac{2}{7}$, we can write $\sin \theta = -\frac{2}{\sqrt{49}} = -\frac{2}{7}$ and $\cos \theta = \frac{7}{\sqrt{49}} = \frac{7}{7} = 1$.

Finding the Value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$

Now that we have found the value of $\cos \theta$, we can find the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$. Since $\cos \theta = 1$, we have $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right] = \boxed{1}$.

Conclusion

In this article, we used a sketch to find the exact value of the expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$. We first understood the properties of the inverse tangent function and then sketched the problem using a graph of the tangent function and the line $y = -\frac{2}{7}$. We then used the Pythagorean identity to find the value of $\cos \theta$ and finally found the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$.

Additional Tips and Tricks

• When using a sketch to find the value of an expression, make sure to consider the properties of the functions involved.
• Use the Pythagorean identity to find the value of $\cos \theta$ when given the value of $\sin \theta$.
• When finding the value of $\cos \left[\tan ^{-1}x\right]$, make sure to consider the range of the inverse tangent function.

Common Mistakes to Avoid

• When sketching the problem, make sure to consider the graph of the tangent function and the line $y = x$.
• When using the Pythagorean identity, make sure to square the values of $\sin \theta$ and $\cos \theta$ before adding them together.
• When finding the value of $\cos \left[\tan ^{-1}x\right]$, make sure to consider the range of the inverse tangent function.

Real-World Applications

• The expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ can be used to model real-world problems involving right triangles and trigonometric functions.
• The Pythagorean identity can be used to find the value of $\cos \theta$ in a variety of real-world applications, such as physics and engineering.

Final Thoughts

In conclusion, using a sketch to find the exact value of the expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ involves understanding the properties of the inverse tangent function and using the Pythagorean identity to find the value of $\cos \theta$. By following these steps and avoiding common mistakes, we can find the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ and apply it to real-world problems.

Frequently Asked Questions

Q: What is the inverse tangent function?

A: The inverse tangent function, denoted as $\tan^{-1}x$, is the angle whose tangent is equal to $x$. In other words, if $\tan \theta = x$, then $\theta = \tan^{-1}x$. The range of the inverse tangent function is typically taken to be $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Q: How do I sketch the problem?

A: To sketch the problem, you need to consider the graph of the tangent function and the line $y = -\frac{2}{7}$. The point of intersection between the graph of the tangent function and the line $y = -\frac{2}{7}$ will give you the value of $\tan^{-1}\left(-\frac{2}{7}\right)$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states $\cos^2 \theta + \sin^2 \theta = 1$. This identity can be used to find the value of $\cos \theta$ when given the value of $\sin \theta$.

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

A: To use the Pythagorean identity to find the value of $\cos \theta$, you need to square the values of $\sin \theta$ and $\cos \theta$ before adding them together. For example, if $\sin \theta = -\frac{2}{7}$, then $\cos^2 \theta = 1 - \left(-\frac{2}{7}\right)^2 = 1 - \frac{4}{49} = \frac{45}{49}$.

Q: What is the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$?

A: The value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ is $\boxed{1}$.

Q: Can I use the Pythagorean identity to find the value of $\cos \theta$ in other real-world applications?

A: Yes, the Pythagorean identity can be used to find the value of $\cos \theta$ in a variety of real-world applications, such as physics and engineering.

Q: What are some common mistakes to avoid when using a sketch to find the value of an expression?

A: Some common mistakes to avoid when using a sketch to find the value of an expression include:

• Not considering the graph of the tangent function and the line $y = x$.
• Not squaring the values of $\sin \theta$ and $\cos \theta$ before adding them together.
• Not considering the range of the inverse tangent function.

Q: Can I use a graphing calculator to find the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$?

A: Yes, you can use a graphing calculator to find the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$. Simply graph the tangent function and the line $y = -\frac{2}{7}$, and then use the calculator to find the value of $\cos \theta$ at the point of intersection.

Q: What are some real-world applications of the expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$?

A: The expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ can be used to model real-world problems involving right triangles and trigonometric functions. Some examples include:

• Physics: The expression can be used to find the value of the cosine of an angle in a right triangle.
• Engineering: The expression can be used to find the value of the cosine of an angle in a right triangle, which is important in the design of buildings and bridges.

Conclusion

In this Q&A article, we have discussed how to use a sketch to find the exact value of the expression $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$. We have also covered some common mistakes to avoid and real-world applications of the expression. By following these steps and avoiding common mistakes, you can find the value of $\cos \left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$ and apply it to real-world problems.