Finding the Adjacent Side and Hypotenuse

Oct 13, 2025 by miodeya.com 36 views

Introduction

In trigonometry, the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Given a point ${$ (3,1)$}$ in a coordinate plane, we can use this definition to find the cosine of the angle formed by the line connecting the origin and the given point.

Understanding the Problem

To find the cosine of the angle, we need to identify the adjacent side and the hypotenuse of the right triangle formed by the given point and the origin. The adjacent side is the side that lies between the angle and the origin, while the hypotenuse is the side opposite the angle.

Visualizing the Problem

Let's visualize the problem by plotting the point ${$ (3,1)$}$ in a coordinate plane. We can see that the point lies in the first quadrant of the plane, and the line connecting the origin and the point forms an angle with the x-axis.

Finding the Adjacent Side and Hypotenuse

To find the adjacent side and the hypotenuse, we can use the distance formula, which is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where ${$ (x_1, y_1)$}$ is the origin, and ${$ (x_2, y_2)$}$ is the given point.

Plugging in the values, we get:

d = \sqrt{(3 - 0)^2 + (1 - 0)^2}

d = \sqrt{9 + 1}

d = \sqrt{10}

So, the length of the hypotenuse is ${$ \sqrt{10}$}$.

Finding the Cosine of the Angle

Now that we have the length of the hypotenuse, we can find the cosine of the angle using the definition:

\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}

In this case, the adjacent side is the x-coordinate of the given point, which is 3.

\cos \theta = \frac{3}{\sqrt{10}}

Simplifying the Expression

To simplify the expression, we can rationalize the denominator by multiplying both the numerator and the denominator by ${$ \sqrt{10}$}$:

\cos \theta = \frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}}

\cos \theta = \frac{3\sqrt{10}}{10}

Conclusion

In this article, we used the definition of cosine of an angle in a right triangle to find the cosine of the angle formed by the line connecting the origin and the point ${$ (3,1)$}$. We identified the adjacent side and the hypotenuse, and used the distance formula to find their lengths. Finally, we simplified the expression for the cosine of the angle.

Applications of Cosine Function

The cosine function has many applications in mathematics and physics. Some of the most common applications include:

Trigonometry: The cosine function is used to solve triangles and find unknown sides and angles.
Physics: The cosine function is used to describe the motion of objects in terms of their position, velocity, and acceleration.
Engineering: The cosine function is used to design and analyze systems that involve periodic motion, such as oscillators and filters.

Real-World Examples

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

Navigation: The cosine function is used in navigation systems to determine the position and orientation of a vehicle or aircraft.
Medical Imaging: The cosine function is used in medical imaging techniques such as MRI and CT scans to reconstruct images of the body.
Signal Processing: The cosine function is used in signal processing techniques such as filtering and modulation to analyze and manipulate signals.

Conclusion

In conclusion, the cosine function is a fundamental concept in mathematics and physics that has many applications in real-world problems. By understanding the definition and properties of the cosine function, we can solve problems and analyze systems that involve periodic motion and trigonometry.

Introduction

In our previous article, we discussed the concept of the cosine function and how it is used to find the cosine of an angle in a right triangle. In this article, we will answer some frequently asked questions about the cosine function.

Q: What is the cosine function?

A: The cosine function is a mathematical function that is used to find the cosine of an angle in a right triangle. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I find the cosine of an angle?

A: To find the cosine of an angle, you need to identify the adjacent side and the hypotenuse of the right triangle formed by the angle. Then, you can use the formula:

\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}

Q: What is the range of the cosine function?

A: The range of the cosine function is [-1, 1]. This means that the cosine of any angle will always be between -1 and 1.

Q: What is the period of the cosine function?

A: The period of the cosine function is 2π. This means that the cosine function repeats itself every 2π radians.

Q: How do I use the cosine function in real-world problems?

A: The cosine function has many real-world applications, including navigation, medical imaging, and signal processing. It is used to describe the motion of objects in terms of their position, velocity, and acceleration.

Q: What are some common mistakes to avoid when using the cosine function?

A: Some common mistakes to avoid when using the cosine function include:

Not identifying the adjacent side and hypotenuse correctly
Not using the correct formula for the cosine function
Not considering the range and period of the cosine function

Q: How do I graph the cosine function?

A: To graph the cosine function, you need to plot the points (x, y) where x is the angle and y is the cosine of the angle. You can use a graphing calculator or software to graph the cosine function.

Q: What are some common applications of the cosine function in physics?

A: Some common applications of the cosine function in physics include:

Motion in a straight line: The cosine function is used to describe the motion of an object in a straight line.
Circular motion: The cosine function is used to describe the motion of an object in a circular path.
Wave motion: The cosine function is used to describe the motion of a wave.

Q: What are some common applications of the cosine function in engineering?

A: Some common applications of the cosine function in engineering include:

Filter design: The cosine function is used to design filters that can remove unwanted frequencies from a signal.
Modulation: The cosine function is used to modulate a signal to change its frequency or amplitude.
Signal processing: The cosine function is used to analyze and manipulate signals in various applications.

Conclusion

In this article, we answered some frequently asked questions about the cosine function. We discussed the definition and properties of the cosine function, and provided examples of its use in real-world problems. We also highlighted some common mistakes to avoid when using the cosine function, and provided information on how to graph the cosine function.