Finding Unknown MeasuresFind The Correct Values For The Variables That Make The Statement $\cos (H)=\frac{x}{y}$ True.Given: $H=60^\circ$Find: $x = \square$ $y = \sqrt{80}$

Introduction

In trigonometry, the cosine function is used to find the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Given the angle $H$ and the ratio $\cos (H)=\frac{x}{y}$, we need to find the values of $x$ and $y$ that make the statement true. In this article, we will use the given angle $H=60^\circ$ to find the correct values of $x$ and $y$.

Understanding the Cosine Function

The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Mathematically, it can be represented as $\cos (H)=\frac{adjacent}{hypotenuse}$. In this case, we are given the ratio $\cos (H)=\frac{x}{y}$, where $x$ is the adjacent side and $y$ is the hypotenuse.

Finding the Value of $x$

To find the value of $x$, we need to use the given angle $H=60^\circ$ and the cosine function. We can start by drawing a right-angled triangle with the angle $H=60^\circ$. Let's assume the length of the adjacent side is $x$ and the length of the hypotenuse is $y$. Using the cosine function, we can write:

$$\cos (H)=\frac{x}{y}$$

Since we are given the angle $H=60^\circ$, we can use the cosine value of $60^\circ$ to find the value of $x$. The cosine value of $60^\circ$ is $\frac{1}{2}$. Therefore, we can write:

$$\frac{1}{2}=\frac{x}{y}$$

Finding the Value of $y$

We are given that $y = \sqrt{80}$. Therefore, we can substitute this value into the equation:

$$\frac{1}{2}=\frac{x}{\sqrt{80}}$$

Solving for $x$

To solve for $x$, we can multiply both sides of the equation by $\sqrt{80}$:

$$x=\frac{1}{2}\times\sqrt{80}$$

Simplifying the expression, we get:

$$x=\frac{\sqrt{80}}{2}$$

Simplifying the Expression

To simplify the expression, we can rewrite $\sqrt{80}$ as $\sqrt{16\times5}$:

$$x=\frac{\sqrt{16\times5}}{2}$$

Simplifying further, we get:

$$x=\frac{4\sqrt{5}}{2}$$

Final Answer

Therefore, the value of $x$ is $\boxed{2\sqrt{5}}$.

Conclusion

In this article, we used the given angle $H=60^\circ$ and the cosine function to find the values of $x$ and $y$ that make the statement $\cos (H)=\frac{x}{y}$ true. We found that the value of $x$ is $2\sqrt{5}$ and the value of $y$ is $\sqrt{80}$.

Additional Information

The cosine function is an important concept in trigonometry, and it has many real-world applications. For example, it is used in navigation, physics, and engineering to find the position and orientation of objects in space.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Mathematics for Engineers and Scientists" by Donald R. Hill

Discussion

This article provides a step-by-step solution to the problem of finding the values of $x$ and $y$ that make the statement $\cos (H)=\frac{x}{y}$ true. The solution uses the given angle $H=60^\circ$ and the cosine function to find the values of $x$ and $y$. The final answer is $x = 2\sqrt{5}$ and $y = \sqrt{80}$.

Introduction

In our previous article, we discussed how to find the values of $x$ and $y$ that make the statement $\cos (H)=\frac{x}{y}$ true. We used the given angle $H=60^\circ$ and the cosine function to find the values of $x$ and $y$. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the cosine function?

A: The cosine function is a mathematical function that is used to find the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Q: How do I use the cosine function to find the values of $x$ and $y$?

A: To use the cosine function to find the values of $x$ and $y$, you need to know the angle $H$ and the ratio $\cos (H)=\frac{x}{y}$. You can then use the cosine value of the angle to find the value of $x$ and the value of $y$.

Q: What is the value of $x$ when $H=60^\circ$?

A: When $H=60^\circ$, the value of $x$ is $2\sqrt{5}$.

Q: What is the value of $y$ when $H=60^\circ$?

A: When $H=60^\circ$, the value of $y$ is $\sqrt{80}$.

Q: How do I simplify the expression $\frac{\sqrt{80}}{2}$?

A: To simplify the expression $\frac{\sqrt{80}}{2}$, you can rewrite $\sqrt{80}$ as $\sqrt{16\times5}$ and then simplify further to get $\frac{4\sqrt{5}}{2}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $x = 2\sqrt{5}$ and $y = \sqrt{80}$.

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

A: The cosine function has many real-world applications, including navigation, physics, and engineering. It is used to find the position and orientation of objects in space.

Q: Can I use the cosine function to find the values of $x$ and $y$ for any angle $H$?

A: Yes, you can use the cosine function to find the values of $x$ and $y$ for any angle $H$. However, you need to know the cosine value of the angle to do so.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the values of $x$ and $y$ that make the statement $\cos (H)=\frac{x}{y}$ true. We provided step-by-step solutions to the problem and discussed some real-world applications of the cosine function.

Additional Information

The cosine function is an important concept in trigonometry, and it has many real-world applications. For example, it is used in navigation, physics, and engineering to find the position and orientation of objects in space.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Mathematics for Engineers and Scientists" by Donald R. Hill

Discussion

This article provides a Q&A section to help readers understand the problem of finding the values of $x$ and $y$ that make the statement $\cos (H)=\frac{x}{y}$ true. The answers to the questions provide step-by-step solutions to the problem and discuss some real-world applications of the cosine function.