Determine The Value Of A Using The Law Of Sines

Introduction

In trigonometry, the Law of Sines is a fundamental concept that establishes a relationship between the sides and angles of a triangle. This law is particularly useful for solving triangles when we have information about some angles and sides, but not enough to directly apply other trigonometric methods like the Law of Cosines or basic trigonometric ratios (SOH-CAH-TOA). In this comprehensive guide, we'll walk through the process of using the Law of Sines to determine the value of a side in a triangle, given some angles and another side. Specifically, we'll focus on a problem where we are given two angles and one side, and our goal is to find the length of another side.

In the realm of trigonometry, the Law of Sines stands as a cornerstone for solving triangles, particularly when dealing with non-right triangles. This law provides a powerful relationship between the angles and sides of any triangle, making it an indispensable tool for various applications in mathematics, physics, and engineering. The beauty of the Law of Sines lies in its ability to connect the ratios of the sines of angles to the lengths of their opposite sides. This connection allows us to solve for unknown sides or angles when we have sufficient information, such as two angles and a side (AAS) or two sides and an angle opposite one of them (SSA). The Law of Sines is expressed mathematically as ${\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}}$, where ${a, b, c}$ are the lengths of the sides of the triangle, and ${A, B, C}$ are the angles opposite those sides, respectively. This formula allows us to set up proportions between the sides and the sines of their opposite angles, which can then be solved using cross-multiplication or other algebraic techniques. One of the most common applications of the Law of Sines is in solving triangles where we are given two angles and a side (AAS). In this scenario, we can use the Law of Sines to find the lengths of the other two sides. Similarly, if we are given two sides and an angle opposite one of them (SSA), we can use the Law of Sines to find the remaining angles and side. However, it's important to note that the SSA case can sometimes lead to ambiguous solutions, where there may be two possible triangles that satisfy the given conditions. In such cases, careful consideration and additional analysis are required to determine the correct solution. The Law of Sines is not only a theoretical concept but also a practical tool with numerous real-world applications. Surveyors use it to calculate distances and angles in land measurement, navigators use it to determine the position and course of ships and aircraft, and engineers use it in structural analysis and design. Its versatility and wide range of applications make it an essential topic in trigonometry and a valuable skill for anyone working with triangles and angles.

1. Understanding the Law of Sines

The Law of Sines is a trigonometric principle that states 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, it's expressed as:

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

Where:

• ${a, b, c}$ are the lengths of the sides of the triangle.
• ${A, B, C}$ are the angles opposite to sides ${a, b, c}$ respectively.

The Law of Sines is a powerful tool in trigonometry, allowing us to solve triangles when we have certain information about their sides and angles. It is particularly useful when dealing with non-right triangles, where the basic trigonometric ratios (SOH-CAH-TOA) cannot be directly applied. The law 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. This relationship can be expressed mathematically as ${\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}}$, where ${a, b, c}$ are the lengths of the sides of the triangle, and ${A, B, C}$ are the angles opposite those sides, respectively. This formula allows us to set up proportions between the sides and the sines of their opposite angles, which can then be solved using cross-multiplication or other algebraic techniques. One of the most common applications of the Law of Sines is in solving triangles where we are given two angles and a side (AAS). In this scenario, we can use the Law of Sines to find the lengths of the other two sides. For example, if we know the lengths of sides ${a}$ and ${b}$, and the angles ${A}$ and ${B}$, we can set up the proportion ${\frac{a}{\sin(A)} = \frac{b}{\sin(B)}}$ and solve for the unknown side. Similarly, if we are given two sides and an angle opposite one of them (SSA), we can use the Law of Sines to find the remaining angles and side. However, it's important to note that the SSA case can sometimes lead to ambiguous solutions, where there may be two possible triangles that satisfy the given conditions. In such cases, careful consideration and additional analysis are required to determine the correct solution. The Law of Sines is not only a theoretical concept but also a practical tool with numerous real-world applications. Surveyors use it to calculate distances and angles in land measurement, navigators use it to determine the position and course of ships and aircraft, and engineers use it in structural analysis and design. Its versatility and wide range of applications make it an essential topic in trigonometry and a valuable skill for anyone working with triangles and angles. Understanding the Law of Sines is crucial for solving a wide range of trigonometric problems. It provides a powerful tool for connecting the sides and angles of a triangle, allowing us to determine unknown values when sufficient information is given.

2. Problem Setup

We are given a triangle with the following information:

• Angle ${A = 45^{\circ}}$
• Angle ${B = 77^{\circ}}$
• Side ${b = 8}$

Our goal is to find the length of side ${a}$, which is opposite to angle ${A}$. To solve this problem, we will use the Law of Sines, which provides a relationship between the sides and angles of a triangle.

Setting up the problem correctly is a critical first step in solving any mathematical problem, especially in trigonometry. In this particular scenario, we are presented with a triangle where we know two angles and one side, and our objective is to determine the length of another side. This type of problem is a classic application of the Law of Sines, which provides a powerful tool for relating the sides and angles of a triangle. Before we can apply the Law of Sines, it's essential to clearly identify the given information and what we are trying to find. We are given that angle ${A = 45^{\circ}}$, angle ${B = 77^{\circ}}$, and side ${b = 8}$. Our goal is to find the length of side ${a}$, which is opposite to angle ${A}$. It's helpful to visualize the triangle and label the sides and angles accordingly. This will make it easier to apply the Law of Sines and set up the correct proportions. Once we have the problem set up correctly, we can proceed to apply 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. This relationship can be expressed mathematically as ${\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}}$, where ${a, b, c}$ are the lengths of the sides of the triangle, and ${A, B, C}$ are the angles opposite those sides, respectively. In our case, we are interested in finding the length of side ${a}$, so we will focus on the proportion involving sides ${a}$ and ${b}$, and angles ${A}$ and ${B}$. This will allow us to set up an equation that we can solve for ${a}$. By carefully setting up the problem and identifying the relevant information, we can ensure that we are on the right track to finding the solution. The next step will be to apply the Law of Sines and solve for the unknown side ${a}$.

3. Applying the Law of Sines

Using the Law of Sines, we can set up the following proportion:

$$\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$$

Substituting the given values:

$$\frac{\sin(45^{\circ})}{a} = \frac{\sin(77^{\circ})}{8}$$

To solve for ${a}$, we can cross-multiply.

Now, let's delve into the process of applying the Law of Sines to the given problem. As we've established, the Law of Sines provides a powerful relationship between the sides and angles of a triangle, allowing us to solve for unknown values when we have sufficient information. In this case, we are given two angles and one side, and our objective is to find the length of another side. To begin, we need to set up the appropriate proportion using 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 ${\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}}$, where ${a, b, c}$ are the lengths of the sides of the triangle, and ${A, B, C}$ are the angles opposite those sides, respectively. In our problem, we are given angle ${A = 45^{\circ}}$, angle ${B = 77^{\circ}}$, and side ${b = 8}$. We want to find the length of side ${a}$, which is opposite to angle ${A}$. Therefore, we will focus on the proportion involving sides ${a}$ and ${b}$, and angles ${A}$ and ${B}$. This allows us to set up the following equation: ${\frac{\sin(A)}{a} = \frac{\sin(B)}{b}}$. Now, we can substitute the given values into this equation. We know that ${A = 45^{\circ}}$, ${B = 77^{\circ}}$, and ${b = 8}$. Plugging these values into the equation, we get: ${\frac{\sin(45^{\circ})}{a} = \frac{\sin(77^{\circ})}{8}}$. This equation sets up a proportion that we can solve for the unknown side ${a}$. To do this, we will use cross-multiplication, which is a technique for solving proportions. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. In this case, we will multiply ${\sin(45^{\circ})}$ by 8 and ${a}$ by ${\sin(77^{\circ})}$. This will give us a new equation that we can solve for ${a}$. By carefully applying the Law of Sines and setting up the correct proportion, we have taken a significant step towards finding the solution. The next step will be to perform the cross-multiplication and solve the resulting equation for ${a}$.

4. Cross-Multiplication

Cross-multiplying gives us:

$$8 \sin(45^{\circ}) = a \sin(77^{\circ})$$

This equation relates the known values to the unknown side ${a}$. We will now isolate ${a}$ to find its value. The process of cross-multiplication is a fundamental technique in solving proportions, and it plays a crucial role in applying the Law of Sines to find unknown side lengths or angles in a triangle. In the previous step, we set up the proportion ${\frac{\sin(45^{\circ})}{a} = \frac{\sin(77^{\circ})}{8}}$ using the Law of Sines, where we are trying to find the value of side ${a}$. To solve for ${a}$, we need to eliminate the fractions in the equation. This is where cross-multiplication comes into play. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. In other words, we multiply the top of the first fraction by the bottom of the second fraction, and we multiply the bottom of the first fraction by the top of the second fraction. Then, we set these two products equal to each other. In our case, we will multiply ${\sin(45^{\circ})}$ by 8 and ${a}$ by ${\sin(77^{\circ})}$. This gives us the equation: ${8 \sin(45^{\circ}) = a \sin(77^{\circ})}$. This equation is a crucial step forward in solving for ${a}$. It eliminates the fractions and sets up a linear equation that we can easily manipulate to isolate ${a}$. The equation ${8 \sin(45^{\circ}) = a \sin(77^{\circ})}$ now relates the known values, which are 8, ${\sin(45^{\circ})}$, and ${\sin(77^{\circ})}$, to the unknown side ${a}$. Our next step is to isolate ${a}$ on one side of the equation. This will allow us to determine its value. To isolate ${a}$, we will need to divide both sides of the equation by ${\sin(77^{\circ})}$. This will cancel out the ${\sin(77^{\circ})}$ on the right side of the equation, leaving us with ${a}$ by itself. The process of cross-multiplication is a fundamental algebraic technique that is widely used in mathematics and other fields. It is a simple yet powerful tool for solving proportions and equations involving fractions. By mastering cross-multiplication, we can confidently tackle a wide range of problems, including those involving the Law of Sines. In the next step, we will proceed to isolate ${a}$ and calculate its value.

5. Isolate 'a'

To isolate ${a}$, divide both sides by ${\sin(77^{\circ})}$:

$$a = \frac{8 \sin(45^{\circ})}{\sin(77^{\circ})}$$

Now we have an expression for ${a}$ in terms of known values. Next, we'll calculate the value of ${a}$.

Isolating 'a' is a critical step in solving for the unknown side length in our trigonometric problem. After applying the Law of Sines and performing cross-multiplication, we arrived at the equation ${8 \sin(45^{\circ}) = a \sin(77^{\circ})}$. Our goal now is to get ${a}$ by itself on one side of the equation so that we can determine its value. To achieve this, we need to perform an algebraic operation that will