Applying the Law of Cosines to the Given Angles

Oct 13, 2025 by pinkandtrash.com 89 views

Introduction

The Law of Cosines is a fundamental concept in trigonometry that allows us to find the length of a side of a triangle when we know the lengths of the other two sides and the angle between them. In this article, we will apply the Law of Cosines to solve triangles with given angles. We will use the formula $a^2 = b^2 + c^2 - 2bc \cos(A)$ , where $a$ , $b$ , and $c$ are the sides of the triangle, and $A$ is the angle between sides $b$ and $c$ .

The Law of Cosines

The Law of Cosines states that for any triangle with sides $a$ , $b$ , and $c$ , and angle $A$ between sides $b$ and $c$ , the following equation holds:

a^2 = b^2 + c^2 - 2bc \cos(A)

This formula can be used to find the length of side $a$ when we know the lengths of sides $b$ and $c$ , and the angle between them.

Applying the Law of Cosines to the Given Angles

We are given three angles: $26^{\circ}$ , $30^{\circ}$ , and $33^{\circ}$ . We can use the Law of Cosines to find the length of the side opposite each of these angles.

Angle $26^{\circ}$

Let's say we have a triangle with angle $26^{\circ}$ , and sides $b$ and $c$ adjacent to this angle. We can use the Law of Cosines to find the length of the side opposite this angle, which we'll call $a$ .

a^2 = b^2 + c^2 - 2bc \cos(26^{\circ})

To solve for $a$ , we need to know the values of $b$ and $c$ . Let's assume that $b = 5$ and $c = 7$ . Then, we can plug these values into the equation:

a^2 = 5^2 + 7^2 - 2(5)(7) \cos(26^{\circ})

Simplifying the equation, we get:

a^2 = 25 + 49 - 70 \cos(26^{\circ})

Using a calculator to evaluate the cosine function, we get:

a^2 = 74 - 70(0.8988)

Simplifying further, we get:

a^2 = 74 - 62.836

a^2 = 11.164

Taking the square root of both sides, we get:

a = \sqrt{11.164}

a = 3.34

Therefore, the length of the side opposite the $26^{\circ}$ angle is approximately $3.34$ units.

Angle $30^{\circ}$

Let's say we have a triangle with angle $30^{\circ}$ , and sides $b$ and $c$ adjacent to this angle. We can use the Law of Cosines to find the length of the side opposite this angle, which we'll call $a$ .

a^2 = b^2 + c^2 - 2bc \cos(30^{\circ})

To solve for $a$ , we need to know the values of $b$ and $c$ . Let's assume that $b = 4$ and $c = 6$ . Then, we can plug these values into the equation:

a^2 = 4^2 + 6^2 - 2(4)(6) \cos(30^{\circ})

Simplifying the equation, we get:

a^2 = 16 + 36 - 48 \cos(30^{\circ})

Using a calculator to evaluate the cosine function, we get:

a^2 = 52 - 48(0.8660)

Simplifying further, we get:

a^2 = 52 - 41.488

a^2 = 10.512

Taking the square root of both sides, we get:

a = \sqrt{10.512}

a = 3.23

Therefore, the length of the side opposite the $30^{\circ}$ angle is approximately $3.23$ units.

Angle $33^{\circ}$

Let's say we have a triangle with angle $33^{\circ}$ , and sides $b$ and $c$ adjacent to this angle. We can use the Law of Cosines to find the length of the side opposite this angle, which we'll call $a$ .

a^2 = b^2 + c^2 - 2bc \cos(33^{\circ})

To solve for $a$ , we need to know the values of $b$ and $c$ . Let's assume that $b = 3$ and $c = 5$ . Then, we can plug these values into the equation:

a^2 = 3^2 + 5^2 - 2(3)(5) \cos(33^{\circ})

Simplifying the equation, we get:

a^2 = 9 + 25 - 30 \cos(33^{\circ})

Using a calculator to evaluate the cosine function, we get:

a^2 = 34 - 30(0.8387)

Simplifying further, we get:

a^2 = 34 - 25.161

a^2 = 8.839

Taking the square root of both sides, we get:

a = \sqrt{8.839}

a = 2.97

Therefore, the length of the side opposite the $33^{\circ}$ angle is approximately $2.97$ units.

Conclusion

In this article, we applied the Law of Cosines to solve triangles with given angles. We used the formula $a^2 = b^2 + c^2 - 2bc \cos(A)$ to find the length of the side opposite each of the given angles. We assumed different values for the sides of the triangle and used a calculator to evaluate the cosine function. The results show that the Law of Cosines can be used to find the length of the side opposite any angle in a triangle, given the lengths of the other two sides and the angle between them.

References

"Law of Cosines" by Math Open Reference. Retrieved from https://www.mathopenref.com/lawofcosines.html
"Trigonometry" by Khan Academy. Retrieved from https://www.khanacademy.org/math/trigonometry

Introduction

In our previous article, we applied the Law of Cosines to solve triangles with given angles. We used the formula $a^2 = b^2 + c^2 - 2bc \cos(A)$ to find the length of the side opposite each of the given angles. In this article, we will answer some common questions related to the Law of Cosines and provide additional examples to help you understand this concept better.

Q: What is the Law of Cosines?

A: The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to find the length of the side opposite a given angle, given the lengths of the other two sides and the angle between them.

Q: When can I use the Law of Cosines?

A: You can use the Law of Cosines when you know the lengths of two sides of a triangle and the angle between them, and you want to find the length of the third side.

Q: How do I apply the Law of Cosines?

A: To apply the Law of Cosines, you need to follow these steps:

Identify the sides of the triangle and the angle between them.
Plug the values of the sides and the angle into the formula $a^2 = b^2 + c^2 - 2bc \cos(A)$ .
Simplify the equation and solve for $a$ .

Q: What if I don't know the value of the cosine function?

A: If you don't know the value of the cosine function, you can use a calculator to evaluate it. Alternatively, you can use a trigonometric table or a calculator with a built-in trigonometric function.

Q: Can I use the Law of Cosines to find the angle between two sides?

A: No, the Law of Cosines is used to find the length of the side opposite a given angle, not the angle between two sides. To find the angle between two sides, you can use the Law of Sines or the Law of Cosines in a different way.

Q: What are some common mistakes to avoid when applying the Law of Cosines?

A: Some common mistakes to avoid when applying the Law of Cosines include:

Not identifying the correct sides of the triangle and the angle between them.
Not plugging the correct values into the formula.
Not simplifying the equation correctly.
Not solving for the correct variable.

Q: Can I use the Law of Cosines to solve right triangles?

A: Yes, you can use the Law of Cosines to solve right triangles. However, in a right triangle, the Law of Cosines can be simplified to the Pythagorean Theorem, which is a more straightforward formula to use.

Q: What are some real-world applications of the Law of Cosines?

A: The Law of Cosines has many real-world applications, including:

Navigation: The Law of Cosines is used in navigation to find the distance between two points on the Earth's surface.
Physics: The Law of Cosines is used in physics to describe the motion of objects in two-dimensional space.
Engineering: The Law of Cosines is used in engineering to design and analyze structures such as bridges and buildings.

Conclusion

In this article, we answered some common questions related to the Law of Cosines and provided additional examples to help you understand this concept better. We hope that this article has been helpful in clarifying any doubts you may have had about the Law of Cosines.

References

"Law of Cosines" by Math Open Reference. Retrieved from https://www.mathopenref.com/lawofcosines.html
"Trigonometry" by Khan Academy. Retrieved from https://www.khanacademy.org/math/trigonometry