Solving For Z Which Equation Correctly Applies The Law Of Sines

In trigonometry, the law of sines is a fundamental concept used to solve triangles, especially when dealing with non-right triangles. It establishes a relationship between the sides of a triangle and the sines of their opposite angles. Understanding the law of sines is crucial for various applications, including navigation, surveying, and engineering. This article delves into the law of sines, its applications, and how it can be used to solve for unknown sides and angles in triangles. We will examine two equations provided and determine which one is correctly set up to solve for the unknown value, z, using the law of sines. The correct application of the law of sines ensures accurate calculations and solutions in trigonometric problems.

Understanding the Law of Sines

The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles in the triangle. Mathematically, this can be expressed as:

rac{a}{\sin(A)} = rac{b}{\sin(B)} = rac{c}{\sin(C)}

Where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite those sides, respectively. This law is particularly useful when you know either two angles and one side (AAS or ASA) or two sides and an angle opposite one of them (SSA). The law of sines allows us to set up proportions that relate the known quantities to the unknown ones, making it a versatile tool in solving triangles. The power of the law of sines lies in its ability to handle both acute and obtuse triangles, providing a consistent method for finding missing dimensions.

Conditions for Using the Law of Sines

The law of sines is applicable under specific conditions, making it essential to identify when it can be used. The primary scenarios include:

1. Angle-Angle-Side (AAS): When two angles and a non-included side are known.
2. Angle-Side-Angle (ASA): When two angles and the included side are known.
3. Side-Side-Angle (SSA): When two sides and an angle opposite one of them are known. This case is also known as the ambiguous case because it may result in zero, one, or two possible triangles.

In each of these scenarios, the law of sines provides a direct method to set up equations and solve for the unknown quantities. Understanding these conditions ensures that you can effectively apply the law of sines to various triangle problems. Recognizing the appropriate conditions also helps in avoiding common pitfalls and misapplications of the law.

Analyzing the Given Equations

We are presented with two equations and must determine which one correctly applies the law of sines to solve for z:

1. $$\frac{\sin(51^{\circ})}{2.6} = \frac{\sin(76^{\circ})}{z}$$

2. $$\frac{\sin(519)}{2.6} = \frac{\sin(53)^{\circ}}{z}$$

To identify the correct equation, we need to ensure that the ratios of the sine of an angle to the length of its opposite side are properly matched. Let's break down each equation to assess its validity. Evaluating the structure and components of each equation will reveal which one adheres to the principles of the law of sines. The process involves verifying the angles and their corresponding sides to ensure a consistent and accurate setup.

Detailed Examination of Equation 1

The first equation is:

$$\frac{\sin(51^{\circ})}{2.6} = \frac{\sin(76^{\circ})}{z}$$

In this equation, we have ${\sin(51^{\circ})}$ divided by 2.6 and ${\sin(76^{\circ})}$ divided by z. If we consider this in the context of the law of sines, it implies that 51 degrees is an angle in the triangle, and 2.6 is the length of the side opposite this angle. Similarly, 76 degrees is another angle, and z is the length of the side opposite this angle. This setup aligns perfectly with the law of sines formula, where the ratio of the sine of an angle to its opposite side is constant. Therefore, this equation appears to be a valid application of the law of sines and a promising candidate for solving for z. The structure of the equation accurately reflects the proportional relationship defined by the law of sines.

Detailed Examination of Equation 2

The second equation is:

$$\frac{\sin(519)}{2.6} = \frac{\sin(53)^{\circ}}{z}$$

Upon closer inspection, we notice a potential issue with the value ${\sin(519)}$. Angles in triangles are typically measured between 0 and 180 degrees. An angle of 519 degrees is significantly larger and does not fit the context of a triangle's internal angles. Additionally, the notation ${\sin(53)^{\circ}}$ is unconventional and suggests a possible typographical error. The correct notation should be ${\sin(53^{\circ})}$. The presence of these inconsistencies raises concerns about the validity of this equation as an application of the law of sines. The irregular angle measurement and notational ambiguity cast doubt on its correctness and usability in solving for z. Therefore, this equation is less likely to be the correct one.

Determining the Correct Equation

After analyzing both equations, it's evident that the first equation:

$$\frac{\sin(51^{\circ})}{2.6} = \frac{\sin(76^{\circ})}{z}$$

is the correct one for solving for z using the law of sines. This equation properly relates the sines of the angles to the lengths of their opposite sides, which is the fundamental principle of the law of sines. The second equation contains an unrealistic angle value (519 degrees) and a notational error, making it an invalid application of the law of sines. By adhering to the principles of the law of sines, we can confidently select the first equation as the correct setup for finding the value of z. This conclusion is based on a careful evaluation of each equation's adherence to trigonometric principles.

Solving for z

Now that we have identified the correct equation, we can proceed to solve for z. The equation is:

$$\frac{\sin(51^{\circ})}{2.6} = \frac{\sin(76^{\circ})}{z}$$

To solve for z, we can use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This gives us:

$$z \cdot \sin(51^{\circ}) = 2.6 \cdot \sin(76^{\circ})$$

Next, we isolate z by dividing both sides of the equation by ${\sin(51^{\circ})}$:

$$z = \frac{2.6 \cdot \sin(76^{\circ})}{\sin(51^{\circ})}$$

Now, we can use a calculator to find the values of ${\sin(76^{\circ})}$ and ${\sin(51^{\circ})}$:

$$\sin(76^{\circ}) \approx 0.9703$$

$$\sin(51^{\circ}) \approx 0.7771$$

Substituting these values into the equation, we get:

$$z \approx \frac{2.6 \cdot 0.9703}{0.7771}$$

$$z \approx \frac{2.52278}{0.7771}$$

$$z \approx 3.246$$

Therefore, the value of z is approximately 3.246. This process demonstrates how the law of sines can be applied to solve for unknown sides in a triangle, providing a precise and reliable method for trigonometric calculations. The solution obtained is consistent with the principles of triangle geometry and the law of sines.

In conclusion, the correct equation to solve for z using the law of sines is:

$$\frac{\sin(51^{\circ})}{2.6} = \frac{\sin(76^{\circ})}{z}$$

We arrived at this conclusion by carefully examining both equations and verifying their adherence to the principles of the law of sines. The incorrect equation contained an invalid angle value and a notational error, making it an unsuitable choice. By applying the law of sines correctly, we were able to set up the appropriate proportion and solve for the unknown side length, z. This demonstrates the importance of understanding the fundamentals of trigonometry and applying them accurately to solve problems. The law of sines is a powerful tool for solving triangles, and its correct application ensures accurate results in various fields, including mathematics, engineering, and navigation.