How Many Possible Triangles Can Be Created If $m \angle A=\frac{\pi}{6}, C=18$, And $a=9$?A. 0 Triangles B. 1 Triangle C. 2 Triangles D. Cannot Be Determined Based On The Given Information

Introduction

In geometry, a triangle is a polygon with three sides and three vertices. The study of triangles is a fundamental aspect of mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. One of the key aspects of triangle geometry is the concept of possible triangles, which refers to the number of unique triangles that can be formed under certain conditions.

In this article, we will explore the problem of determining the number of possible triangles that can be created if $m \angle A=\frac{\pi}{6}, c=18$, and $a=9$. This problem is a classic example of a geometry problem that requires the application of trigonometric concepts and theorems.

Understanding the Problem

To solve this problem, we need to understand the given information and the conditions that must be satisfied for a triangle to be formed. The given information includes:

• $m \angle A=\frac{\pi}{6}$: This means that the measure of angle A is $\frac{\pi}{6}$ radians, which is equivalent to 30 degrees.
• $c=18$: This means that the length of side c is 18 units.
• $a=9$: This means that the length of side a is 9 units.

We need to determine the number of possible triangles that can be formed under these conditions.

Applying Trigonometric Concepts

To solve this problem, we can apply the Law of Sines, which states that for any triangle with sides 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 this equation to relate the given information and determine the number of possible triangles.

Using the Law of Sines

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

$$\frac{a}{\sin A} = \frac{c}{\sin C}$$

Substituting the given values, we get:

$$\frac{9}{\sin \frac{\pi}{6}} = \frac{18}{\sin C}$$

Simplifying the equation, we get:

$$\frac{9}{\frac{1}{2}} = \frac{18}{\sin C}$$

$$18 = \frac{18}{\sin C}$$

$$\sin C = 1$$

This means that the sine of angle C is equal to 1, which is only possible if angle C is a right angle (90 degrees).

Conclusion

Based on the analysis above, we can conclude that the number of possible triangles that can be created if $m \angle A=\frac{\pi}{6}, c=18$, and $a=9$ is 1. This is because the sine of angle C is equal to 1, which means that angle C is a right angle, and only one triangle can be formed under these conditions.

The final answer is B. 1 triangle.

Introduction

In our previous article, we explored the problem of determining the number of possible triangles that can be created if $m \angle A=\frac{\pi}{6}, c=18$, and $a=9$. We applied the Law of Sines and concluded that the number of possible triangles is 1. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the significance of the angle A being $\frac{\pi}{6}$ radians?

A: The angle A being $\frac{\pi}{6}$ radians is significant because it is a special angle that has a known sine value. The sine of $\frac{\pi}{6}$ radians is $\frac{1}{2}$, which is a fundamental value in trigonometry.

Q: Why is the sine of angle C equal to 1?

A: The sine of angle C is equal to 1 because the sine function has a maximum value of 1, which occurs when the angle is a right angle (90 degrees). In this case, the sine of angle C is equal to 1 because angle C is a right angle.

Q: Can we form multiple triangles with the same side lengths and angles?

A: No, we cannot form multiple triangles with the same side lengths and angles. The Law of Sines states that for any triangle with sides a, b, and c, and angles A, B, and C, respectively, the following equation holds:

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

This means that the ratio of the side lengths to the sines of the angles is unique for each triangle.

Q: What if we change the value of side c to 19? How many possible triangles can be created?

A: If we change the value of side c to 19, we can no longer form a triangle with the same side lengths and angles. The sine of angle C is no longer equal to 1, and the triangle is no longer a right triangle. In this case, we cannot determine the number of possible triangles that can be created.

Q: Can we use other trigonometric functions to solve this problem?

A: Yes, we can use other trigonometric functions to solve this problem. For example, we can use the cosine function to relate the side lengths and angles of the triangle. However, the sine function is the most convenient function to use in this case because the sine of angle C is equal to 1.

Q: What is the relationship between the number of possible triangles and the value of side c?

A: The number of possible triangles is related to the value of side c in the sense that if side c is too large, we cannot form a triangle with the same side lengths and angles. In this case, we cannot determine the number of possible triangles that can be created.

Conclusion

In this article, we answered some frequently asked questions related to the problem of determining the number of possible triangles that can be created if $m \angle A=\frac{\pi}{6}, c=18$, and $a=9$. We hope that this article has provided a better understanding of the problem and its solution.

Additional Resources

For more information on trigonometry and the Law of Sines, please refer to the following resources:

• Trigonometry Wikipedia Article
• Law of Sines Wikipedia Article
• Trigonometry Textbook

We hope that this article has been helpful in understanding the problem and its solution. If you have any further questions or concerns, please do not hesitate to contact us.