Given $\tan A = \frac{8}{2}$Find $\cos B =$Leave Your Answer In Simplified Radical Form.$\square$

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically the equation $\tan A = \frac{8}{2}$. We will also explore the concept of cosine and how to find $\cos B$ in simplified radical form.

Understanding Trigonometric Functions

Before we dive into solving the equation, let's briefly review the basic trigonometric functions. The three main trigonometric functions are:

• Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Solving the Equation

Now that we have a basic understanding of trigonometric functions, let's solve the equation $\tan A = \frac{8}{2}$. To do this, we need to find the value of $A$.

$\tan A = \frac{8}{2}$

We can simplify the equation by dividing both sides by 2:

$\tan A = 4$

To find the value of $A$, we can use the inverse tangent function, denoted as $\tan^{-1}$. The inverse tangent function returns the angle whose tangent is a given number.

$A = \tan^{-1}(4)$

Using a calculator, we can find the value of $A$:

$A \approx 76.0^{\circ}$

Finding $\cos B$

Now that we have found the value of $A$, let's find $\cos B$. To do this, we need to use the trigonometric identity:

$\cos B = \cos (180^{\circ} - A)$

We can substitute the value of $A$ into the equation:

$\cos B = \cos (180^{\circ} - 76.0^{\circ})$

$\cos B = \cos 104.0^{\circ}$

Using a calculator, we can find the value of $\cos B$:

$\cos B \approx 0.271$

However, we are asked to leave the answer in simplified radical form. To do this, we need to use the trigonometric identity:

$\cos B = \cos (180^{\circ} - A)$

We can substitute the value of $A$ into the equation:

$\cos B = \cos (180^{\circ} - \tan^{-1}(4))$

Using the trigonometric identity:

$\cos (180^{\circ} - \theta) = -\cos \theta$

We can rewrite the equation:

$\cos B = -\cos (\tan^{-1}(4))$

Using the trigonometric identity:

$\cos (\tan^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}$

We can rewrite the equation:

$\cos B = -\frac{1}{\sqrt{4^2 + 1}}$

$\cos B = -\frac{1}{\sqrt{17}}$

$\cos B = -\frac{1}{\sqrt{17}} \times \frac{\sqrt{17}}{\sqrt{17}}$

$\cos B = -\frac{\sqrt{17}}{17}$

Conclusion

In this article, we have solved the equation $\tan A = \frac{8}{2}$ and found the value of $A$. We have also found $\cos B$ in simplified radical form. The final answer is:

$\boxed{-\frac{\sqrt{17}}{17}}$

References

• [1] "Trigonometry" by Michael Corral. Available at: https://math.furman.edu/~mcorral/Trigonometry/Trigonometry.pdf
• [2] "Trigonometric Functions" by Khan Academy. Available at: https://www.khanacademy.org/math/trigonometry

Glossary

• Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
• Inverse Tangent Function: A function that returns the angle whose tangent is a given number.
• Trigonometric Identity: A mathematical statement that relates the trigonometric functions to each other.
  Frequently Asked Questions: Trigonometry and Beyond =====================================================

Q: What is trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation.

Q: What are the basic trigonometric functions?

A: The three main trigonometric functions are:

• Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the variable (usually the angle) using the inverse trigonometric functions. For example, to solve the equation $\tan A = \frac{8}{2}$, you would use the inverse tangent function, denoted as $\tan^{-1}$.

Q: What is the inverse tangent function?

A: The inverse tangent function, denoted as $\tan^{-1}$, returns the angle whose tangent is a given number. For example, $\tan^{-1}(4)$ returns the angle whose tangent is 4.

Q: How do I find $\cos B$ in simplified radical form?

A: To find $\cos B$ in simplified radical form, you need to use the trigonometric identity:

$\cos B = \cos (180^{\circ} - A)$

You can then substitute the value of $A$ into the equation and simplify using the trigonometric identities.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\cos (180^{\circ} - \theta) = -\cos \theta$
• $\cos (\tan^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}$
• $\sin (\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}}$

Q: How do I use trigonometry in real-life applications?

A: Trigonometry has numerous applications in various fields, including:

• Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometry is used to design and build structures, such as bridges and buildings.
• Navigation: Trigonometry is used to determine the position and direction of objects, such as ships and planes.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not using the correct inverse trigonometric function: Make sure to use the correct inverse trigonometric function, such as $\tan^{-1}$ or $\sin^{-1}$.
• Not simplifying the equation: Make sure to simplify the equation using the trigonometric identities.
• Not checking the domain and range: Make sure to check the domain and range of the trigonometric function to ensure that the solution is valid.

Conclusion

In this article, we have answered some frequently asked questions about trigonometry and beyond. We have covered topics such as the basic trigonometric functions, solving trigonometric equations, and using trigonometry in real-life applications. We hope that this article has been helpful in answering your questions and providing you with a better understanding of trigonometry.