Which Equation Correctly Uses The Law Of Cosines To Solve For The Length S S S ?A. 9^2 = S^2 + 10^2 - 2(s)(10) \cos \left(100^{\circ}\right ]B. 9 = S + 10 - 2(s)(10) \cos \left(100^{\circ}\right ]C. $10^2 = S^2 + 100 -

Mar 11, 2025 by ADMIN 219 views

Which Equation Correctly Uses the Law of Cosines to Solve for the Length $s$ ?

The law of cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a powerful tool for solving problems involving triangles, and it has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the law of cosines and examine three different equations to determine which one correctly uses the law of cosines to solve for the length $s$ .

Understanding the Law of Cosines

The law of cosines states that for any triangle with sides of length $a$ , $b$ , and $c$ , and an angle $C$ opposite side $c$ , the following equation holds:

c^2 = a^2 + b^2 - 2ab \cos C

This equation relates the lengths of the sides of the triangle to the cosine of one of its angles. It is a fundamental concept in trigonometry and has numerous applications in various fields.

Examining the Equations

Now, let's examine the three equations given in the problem:

A. $9^2 = s^2 + 10^2 - 2(s)(10) \cos \left(100^{\circ}\right)$

B. $9 = s + 10 - 2(s)(10) \cos \left(100^{\circ}\right)$

C. $10^2 = s^2 + 100 - 2(s)(10) \cos \left(100^{\circ}\right)$

To determine which equation correctly uses the law of cosines, we need to analyze each equation and compare it to the law of cosines equation.

Equation A

Equation A is:

9^2 = s^2 + 10^2 - 2(s)(10) \cos \left(100^{\circ}\right)

This equation appears to be a correct application of the law of cosines. The left-hand side of the equation is the square of the length of side $c$ , which is $9$ . The right-hand side of the equation is the sum of the squares of the lengths of sides $a$ and $b$ , minus twice the product of the lengths of sides $a$ and $b$ times the cosine of angle $C$ . In this case, angle $C$ is $100^{\circ}$ , and the lengths of sides $a$ and $b$ are $s$ and $10$ , respectively.

Equation B

Equation B is:

9 = s + 10 - 2(s)(10) \cos \left(100^{\circ}\right)

This equation does not appear to be a correct application of the law of cosines. The left-hand side of the equation is a single value, $9$ , while the right-hand side of the equation is an expression involving the lengths of sides $s$ and $10$ , as well as the cosine of angle $C$ . This equation does not have the same form as the law of cosines equation, and it does not appear to be a correct solution for the length $s$ .

Equation C

Equation C is:

10^2 = s^2 + 100 - 2(s)(10) \cos \left(100^{\circ}\right)

This equation does not appear to be a correct application of the law of cosines. The left-hand side of the equation is the square of the length of side $c$ , which is $10$ . However, the right-hand side of the equation is not the sum of the squares of the lengths of sides $a$ and $b$ , minus twice the product of the lengths of sides $a$ and $b$ times the cosine of angle $C$ . Instead, the right-hand side of the equation is an expression involving the square of the length of side $s$ , as well as the square of the length of side $10$ , minus twice the product of the lengths of sides $s$ and $10$ times the cosine of angle $C$ . This equation does not have the same form as the law of cosines equation, and it does not appear to be a correct solution for the length $s$ .

Conclusion

Based on our analysis of the three equations, we can conclude that Equation A is the correct application of the law of cosines to solve for the length $s$ . This equation has the same form as the law of cosines equation, and it correctly relates the lengths of the sides of the triangle to the cosine of one of its angles. Therefore, the correct answer is:

The final answer is Equation A.

Additional Examples and Applications

The law of cosines has numerous applications in various fields, including physics, engineering, and navigation. Here are a few additional examples and applications of the law of cosines:

Physics: The law of cosines can be used to calculate the distance between two objects in a two-dimensional space. For example, if we know the distance between two objects and the angle between them, we can use the law of cosines to calculate the distance between the two objects.
Engineering: The law of cosines can be used to calculate the length of a side of a triangle in a two-dimensional space. For example, if we know the lengths of two sides of a triangle and the angle between them, we can use the law of cosines to calculate the length of the third side.
Navigation: The law of cosines can be used to calculate the distance between two points on the surface of the Earth. For example, if we know the latitude and longitude of two points, we can use the law of cosines to calculate the distance between them.

Common Mistakes and Misconceptions

There are several common mistakes and misconceptions that people make when using the law of cosines. Here are a few examples:

Incorrect application of the law of cosines: Many people incorrectly apply the law of cosines by using the wrong formula or by not using the correct values for the lengths of the sides and the angle.
Failure to check units: Many people fail to check the units of the values they are using in the law of cosines equation. This can lead to incorrect results.
Failure to consider the context: Many people fail to consider the context of the problem they are trying to solve. For example, if they are trying to calculate the distance between two points on the surface of the Earth, they may need to consider the curvature of the Earth.

Conclusion

In conclusion, the law of cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a powerful tool for solving problems involving triangles, and it has numerous applications in various fields. By understanding the law of cosines and how to apply it correctly, we can solve a wide range of problems involving triangles.
Q&A: The Law of Cosines

In our previous article, we explored the law of cosines and examined three different equations to determine which one correctly uses the law of cosines to solve for the length $s$ . In this article, we will answer some frequently asked questions about the law of cosines and provide additional examples and applications.

Q: What is the law of cosines?

A: The law of cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a powerful tool for solving problems involving triangles, and it has numerous applications in various fields.

Q: How do I apply the law of cosines?

A: To apply the law of cosines, you need to know the lengths of two sides of a triangle and the angle between them. You can then use the law of cosines equation to calculate the length of the third side.

Q: What is the law of cosines equation?

A: The law of cosines equation is:

c^2 = a^2 + b^2 - 2ab \cos C

where $c$ is the length of the side opposite angle $C$ , $a$ and $b$ are the lengths of the other two sides, and $C$ is the angle between sides $a$ and $b$ .

Q: How do I use the law of cosines to solve for the length of a side?

A: To use the law of cosines to solve for the length of a side, you need to know the lengths of the other two sides and the angle between them. You can then plug these values into the law of cosines equation and solve for the length of the third side.

Q: What are some common mistakes and misconceptions about the law of cosines?

A: Some common mistakes and misconceptions about the law of cosines include:

Incorrect application of the law of cosines
Failure to check units
Failure to consider the context of the problem

Q: How do I check my work when using the law of cosines?

A: To check your work when using the law of cosines, you should:

Verify that you have used the correct formula
Check that you have used the correct values for the lengths of the sides and the angle
Check that your units are correct
Consider the context of the problem and make sure that your solution is reasonable

Q: What are some real-world applications of the law of cosines?

A: Some real-world applications of the law of cosines include:

Physics: The law of cosines can be used to calculate the distance between two objects in a two-dimensional space.
Engineering: The law of cosines can be used to calculate the length of a side of a triangle in a two-dimensional space.
Navigation: The law of cosines can be used to calculate the distance between two points on the surface of the Earth.

Q: How do I use the law of cosines to solve problems involving right triangles?

A: To use the law of cosines to solve problems involving right triangles, you can use the fact that the cosine of a right angle is 0. This means that the law of cosines equation simplifies to:

c^2 = a^2 + b^2

where $c$ is the length of the hypotenuse, $a$ and $b$ are the lengths of the other two sides, and $C$ is the right angle.

Q: How do I use the law of cosines to solve problems involving obtuse triangles?

A: To use the law of cosines to solve problems involving obtuse triangles, you can use the fact that the cosine of an obtuse angle is negative. This means that the law of cosines equation becomes:

c^2 = a^2 + b^2 + 2ab \cos C

where $c$ is the length of the side opposite angle $C$ , $a$ and $b$ are the lengths of the other two sides, and $C$ is the obtuse angle.

Conclusion