Determining the Existence of a Triangle

Oct 13, 2025 by leolavanderia.com.br 27 views

Introduction

In trigonometry, determining the existence and solving triangles is a crucial aspect of understanding the relationships between the sides and angles of a triangle. Given two sides and an angle, we can use various methods to determine whether the given information results in one triangle, two triangles, or no triangle at all. In this article, we will explore the process of determining the existence and solving triangles using the given information.

The Law of Sines

The Law of Sines is a fundamental concept in trigonometry that relates the sides and angles of a triangle. It states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively, the following equation holds:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

We can use the Law of Sines to determine whether the given information results in one triangle, two triangles, or no triangle at all.

Given Information

We are given two sides and an angle: $b = 8$ , $c = 10$ , and $B = 40^{\circ}$ . Our goal is to determine whether the given information results in one triangle, two triangles, or no triangle at all.

Applying the Law of Sines

Using the Law of Sines, we can write the following equation:

\frac{b}{\sin B} = \frac{c}{\sin C}

Substituting the given values, we get:

\frac{8}{\sin 40^{\circ}} = \frac{10}{\sin C}

Simplifying the equation, we get:

\sin C = \frac{10 \sin 40^{\circ}}{8}

Determining the Existence of a Triangle

To determine whether the given information results in one triangle, two triangles, or no triangle at all, we need to check if the calculated value of $\sin C$ is within the range of possible values for $\sin C$ . Since $\sin C$ must be between 0 and 1, we can check if the calculated value is within this range.

Calculating the Value of $\sin C$

Using a calculator, we can calculate the value of $\sin C$ :

\sin C = \frac{10 \sin 40^{\circ}}{8} \approx 0.625

Since the calculated value of $\sin C$ is within the range of possible values for $\sin C$ , we can conclude that the given information results in one triangle.

Solving the Triangle

Now that we have determined that the given information results in one triangle, we can use the Law of Sines to solve the triangle. We can use the following equation to find the length of side a:

\frac{a}{\sin A} = \frac{b}{\sin B}

Substituting the given values, we get:

\frac{a}{\sin A} = \frac{8}{\sin 40^{\circ}}

Simplifying the equation, we get:

a = \frac{8 \sin A}{\sin 40^{\circ}}

Finding the Measure of Angle A

To find the measure of angle A, we can use the fact that the sum of the measures of the angles in a triangle is 180°. We can write the following equation:

A + B + C = 180^{\circ}

Substituting the given values, we get:

A + 40^{\circ} + C = 180^{\circ}

Simplifying the equation, we get:

A + C = 140^{\circ}

Finding the Measure of Angle C

We already calculated the value of $\sin C$ :

\sin C = \frac{10 \sin 40^{\circ}}{8} \approx 0.625

Using a calculator, we can find the measure of angle C:

C \approx \sin^{-1} (0.625) \approx 39.23^{\circ}

Finding the Measure of Angle A

Now that we have found the measure of angle C, we can find the measure of angle A:

A + C = 140^{\circ}

Substituting the value of C, we get:

A + 39.23^{\circ} = 140^{\circ}

Simplifying the equation, we get:

A \approx 100.77^{\circ}

Finding the Length of Side a

Now that we have found the measure of angle A, we can find the length of side a:

a = \frac{8 \sin A}{\sin 40^{\circ}}

Substituting the value of A, we get:

a \approx \frac{8 \sin 100.77^{\circ}}{\sin 40^{\circ}} \approx 12.95

Conclusion

In this article, we used the Law of Sines to determine whether the given information results in one triangle, two triangles, or no triangle at all. We found that the given information results in one triangle and solved the triangle using the Law of Sines. We found the measure of angle A, the measure of angle C, and the length of side a.

Final Answer

The final answer is one triangle with the following measures:

Angle A: $100.77^{\circ}$
Angle C: $39.23^{\circ}$
Side a: $12.95$

Note: The final answer is an approximation, as the values of the angles and the length of side a are calculated using a calculator.

Introduction

In our previous article, we explored the process of determining the existence and solving triangles using the given information. We used the Law of Sines to determine whether the given information results in one triangle, two triangles, or no triangle at all. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the Law of Sines?

A: The Law of Sines is a fundamental concept in trigonometry that relates the sides and angles of a triangle. It states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively, the following equation holds:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Q: How do I determine whether the given information results in one triangle, two triangles, or no triangle at all?

A: To determine whether the given information results in one triangle, two triangles, or no triangle at all, you need to check if the calculated value of $\sin C$ is within the range of possible values for $\sin C$ . Since $\sin C$ must be between 0 and 1, you can check if the calculated value is within this range.

Q: What if the calculated value of $\sin C$ is not within the range of possible values for $\sin C$ ?

A: If the calculated value of $\sin C$ is not within the range of possible values for $\sin C$ , it means that the given information does not result in a valid triangle. In this case, you can conclude that the given information results in no triangle at all.

Q: How do I solve a triangle using the Law of Sines?

A: To solve a triangle using the Law of Sines, you need to use the following equation:

\frac{a}{\sin A} = \frac{b}{\sin B}

Substituting the given values, you can find the length of side a. You can also use the fact that the sum of the measures of the angles in a triangle is 180° to find the measure of angle A.

Q: What if I have two possible values for the measure of angle A?

A: If you have two possible values for the measure of angle A, it means that the given information results in two possible triangles. In this case, you can use the Law of Sines to find the length of side a for each possible triangle.

Q: Can I use the Law of Sines to solve a triangle with three sides and no angles?

A: No, you cannot use the Law of Sines to solve a triangle with three sides and no angles. The Law of Sines requires that you have at least one angle and two sides to solve a triangle.

Q: What if I have a triangle with two sides and an angle, but the angle is not one of the given angles?

A: If you have a triangle with two sides and an angle, but the angle is not one of the given angles, you can use the Law of Sines to find the measure of the given angle. You can then use the fact that the sum of the measures of the angles in a triangle is 180° to find the measure of the other angles.

Conclusion

In this article, we answered some frequently asked questions related to determining the existence and solving triangles using the given information. We hope that this article has provided you with a better understanding of the process and has helped you to solve triangles using the Law of Sines.

Final Answer

The final answer is that the Law of Sines is a powerful tool for determining the existence and solving triangles using the given information. By understanding the Law of Sines and how to apply it, you can solve triangles with ease and accuracy.

Note: The final answer is a summary of the article, and it is not a numerical answer.