Law of Sines: Videos & Practice Problems

To solve non-right triangles, the Law of Sines is essential. It states that the ratio of the sine of an angle to its opposite side is constant across all angles: $(\frac{\sin \left(A\right)}{a}=\frac{\sin \left(B\right)}{b}=\frac{\sin \left(C\right)}{c})$. This method allows for solving for missing sides and angles, particularly in SSA, SAA, and ASA configurations, which can yield one, two, or no solutions, highlighting the concept of ambiguous cases in triangle solving.

Intro to Law of Sines

In solving triangles, understanding the differences between right triangles and non-right triangles is crucial. For right triangles, the Pythagorean theorem and trigonometric ratios (SOHCAHTOA) can be effectively used to find missing sides. However, when dealing with non-right triangles, these methods are not applicable. Instead, the Law of Sines provides a powerful alternative for solving for unknown sides and angles.

The Law of Sines states that the ratio of the sine of an angle to the length of the opposite side is constant for all three angles and sides in a triangle. This can be expressed mathematically as:

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

In this equation, \(A\), \(B\), and \(C\) represent the angles of the triangle, while \(a\), \(b\), and \(c\) are the lengths of the sides opposite those angles, respectively. When using the Law of Sines, it is essential to have at least one complete ratio, meaning you need to know three out of the four variables involved.

To solve for a missing side, you can select two ratios from the Law of Sines that include the known values. For instance, if you know angles \(A\) and \(C\) and side \(c\), you can set up the equation:

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

From here, you can rearrange the equation to isolate the unknown side \(a\):

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

By substituting the known values into this equation, you can calculate the length of side \(a\). For example, if \(c = 6\), \(C = 70^\circ\), and \(A = 30^\circ\), the calculation would be:

$$a = \frac{6 \cdot \sin(30^\circ)}{\sin(70^\circ)}$$

After performing the calculations, you would find that \(a \approx 3.19\). This result aligns with the expectations based on the triangle's dimensions, illustrating how the Law of Sines can effectively be used to solve for unknown sides in non-right triangles.

It's also worth noting that in some contexts, the angles may be represented by Greek letters such as alpha, beta, and gamma, but the application of the Law of Sines remains the same. Mastering this method is essential for tackling a variety of problems involving non-right triangles.

Use the Law of Sines to find the length of side $a$ to two decimal places.

• A. 8.20
• B. 4.39
• C. 2.20
• D. 1.61

Answer: (B)

Use the Law of Sines to find the angle $B$ to the nearest tenth of a degree.

• A. 48.6°
• B. 77.2°
• C. 40.5°
• D. 35.3°

Answer: (A)

Solving SAA & ASA Triangles

To solve for missing sides and angles in non-right triangles, the law of sines is a powerful tool. This method is particularly useful when dealing with specific configurations of triangles, categorized by the information provided at the start of a problem. The three main types are Angle-Side-Angle (ASA), Side-Angle-Angle (SAA), and Side-Side-Angle (SSA). In this context, we will focus on ASA and SAA triangles.

In an ASA triangle, you are given two angles and the side between them. For example, if you know angles A and C, along with side b, you can easily classify the triangle. In an SAA triangle, you have two angles and a side that is adjacent to them, such as angles A and C with side c. The SSA configuration, while common, will not be covered in this discussion.

To classify a triangle, start by sketching it based on the given angles and sides. For instance, if you have angle A = 30 degrees, angle C = 70 degrees, and side c = 6, you can visualize the triangle and confirm it is an SAA triangle. The angles in any triangle must sum to 180 degrees, which can be calculated using the angle sum formula:

$$A + B + C = 180$$

From this, you can find the missing angle B by rearranging the formula:

$$B = 180 - A - C$$

Substituting the known values gives:

$$B = 180 - 30 - 70 = 80$$

Now that all angles are known, you can use the law of sines to find the missing sides. The law of sines states:

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

To find side a, you can set up the equation using the known values:

$$\frac{a}{\sin(30)} = \frac{6}{\sin(70)}$$

Cross-multiplying and solving for a gives:

$$a = \frac{6 \cdot \sin(30)}{\sin(70)}$$

Calculating this results in approximately 3.19 for side a. Next, to find side b, you can use a similar approach:

$$\frac{b}{\sin(80)} = \frac{6}{\sin(70)}$$

Cross-multiplying and solving for b yields:

$$b = \frac{6 \cdot \sin(80)}{\sin(70)}$$

This calculation results in approximately 6.29 for side b. By following these steps—sketching the triangle, applying the angle sum formula, and utilizing the law of sines—you can effectively solve for all missing sides and angles in SAA and ASA triangles.

Classify the triangle, then solve: $A=60°,B=15°,c=6$.

• A. $SAA,a=6.69,b=22.4,C=105\degree$
• B. $ASA,a=6.69,b=22.4,C=105\degree$
• C. $ASA,a=5.38,b=1.61,C=105\degree$
• D. $SAA,a=5.38,b=1.61,C=105\degree$

Answer: (C)

An engineer wants to measure the distance to cross a river. If $B=30\degree$, $a=300$$ft$, $C=100\degree$ find the shortest distance (in $ft$) you’d have to travel to cross the river.

• A. $459.6ft$
• B. $195.8 ft$
• C. $152.3ft$
• D. $233.4ft$

Answer: (B)

Solving SSA Triangles ("Ambiguous" Case)

When solving SSA (Side-Side-Angle) triangles using the law of sines, it's important to recognize that this scenario can lead to three different outcomes: no solution, one solution, or two solutions. This ambiguity arises because the angle given is not between the two known sides, making it less straightforward than other triangle types.

To begin solving an SSA triangle, start by applying the law of sines, which states:

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

In this equation, \(A\) and \(B\) are the angles opposite sides \(a\) and \(b\), respectively. For example, if you know \(a = 6\), \(b = 8\), and \(A = 41^\circ\), you can set up the equation to find \(\sin B\):

$$\sin B = \frac{b \cdot \sin A}{a} = \frac{8 \cdot \sin(41^\circ)}{6}$$

Calculating this gives \(\sin B \approx 0.875\). Since this value is less than 1, we can proceed to find angle \(B\) using the inverse sine function:

$$B = \sin^{-1}(0.875) \approx 61^\circ$$

However, the sine function is positive in both the first and second quadrants, which means there is a second possible angle:

$$B_2 = 180^\circ - 61^\circ = 119^\circ$$

Next, evaluate the feasibility of these angles. For the first angle \(B_1 = 61^\circ\), the sum of angles \(A\) and \(B_1\) is:

$$A + B_1 = 41^\circ + 61^\circ = 102^\circ$$

This sum is less than \(180^\circ\), allowing for a valid triangle. Now, calculate the third angle \(C_1\):

$$C_1 = 180^\circ - (A + B_1) = 180^\circ - 102^\circ = 78^\circ$$

For the second angle \(B_2 = 119^\circ\), the sum is:

$$A + B_2 = 41^\circ + 119^\circ = 160^\circ$$

Since this sum is also less than \(180^\circ\), it is valid. Calculate the third angle \(C_2\):

$$C_2 = 180^\circ - (A + B_2) = 180^\circ - 160^\circ = 20^\circ$$

In summary, the SSA triangle can yield two distinct triangles based on the calculations:

• Triangle 1: Angles \(A = 41^\circ\), \(B_1 = 61^\circ\), \(C_1 = 78^\circ\)
• Triangle 2: Angles \(A = 41^\circ\), \(B_2 = 119^\circ\), \(C_2 = 20^\circ\)

Both configurations are valid solutions to the original problem, illustrating the concept of the ambiguous case in SSA triangles. Understanding this process is crucial for effectively solving such problems in trigonometry.

Solving SSA Triangles ("Ambiguous" Case) Example 1

In this scenario, we are dealing with a triangle where two sides and a non-included angle are given: side \( a = 1 \), side \( b = 4 \), and angle \( A = 30^\circ \). This configuration is known as the Side-Side-Angle (SSA) case, which can lead to ambiguous situations in triangle construction.

To analyze this triangle, we can apply the Law of Sines, which states that the ratio of a side length to the sine of its opposite angle is constant across all sides of the triangle. The formula can be expressed as:

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

In our case, we can rearrange this to find angle \( B \):

\[\sin B = \frac{b \cdot \sin A}{a}\]

Substituting the known values:

\[\sin B = \frac{4 \cdot \sin(30^\circ)}{1}\]

Since \( \sin(30^\circ) = \frac{1}{2} \), we have:

\[\sin B = \frac{4 \cdot \frac{1}{2}}{1} = 2\]

However, the sine function has a range of values between -1 and 1. Since \( \sin B = 2 \) exceeds this range, it indicates that no valid angle \( B \) can exist, leading us to conclude that there is no solution for this triangle.

To visualize this, consider that angle \( A \) is \( 30^\circ \) and side \( b \) is \( 4 \). The side \( a \), which is \( 1 \), is too short to connect to the endpoint of side \( b \) when extended from angle \( A \). Regardless of how angle \( A \) is positioned, side \( a \) cannot reach the necessary length to form a triangle, confirming that a triangle cannot be constructed with the given dimensions.

Thus, the final conclusion is that there is no solution for this triangle configuration due to the limitations imposed by the lengths of the sides and the angle provided.

Solving SSA Triangles ("Ambiguous" Case) Example 2

In this example, we explore a triangle where two sides, denoted as \( b \) and \( c \), and one angle, \( B \), are known. This scenario falls under the Side-Side-Angle (SSA) condition, which allows us to apply the Law of Sines to find additional angles. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant across the triangle, expressed as:

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

In this case, we can ignore side \( a \) since we have no information about it. By rearranging the equation, we can solve for \( \sin C \) as follows:

\[\sin C = \frac{c \cdot \sin B}{b}\]

Substituting the known values, we find:

\[\sin C = \frac{2 \cdot \sin(29^\circ)}{4}\]

Calculating this gives us approximately \( 0.242 \). Since this value is less than 1, we can proceed to find angle \( C \) using the inverse sine function:

\[C_1 = \sin^{-1}(0.242) \approx 14^\circ\]

However, the sine function is positive in both the first and second quadrants, leading to a second possible angle:

\[C_2 = 180^\circ - C_1 = 180^\circ - 14^\circ = 166^\circ\]

Next, we need to determine which of these angles is valid in the context of the triangle. We add \( C_2 \) to the known angle \( B \) (29 degrees) to check if the sum exceeds 180 degrees:

\[B + C_2 = 29^\circ + 166^\circ = 195^\circ\]

This sum is greater than 180 degrees, which is not possible for a triangle. Therefore, \( C_2 \) is invalid, leaving us with only \( C_1 = 14^\circ \) as the feasible angle.

Now, we can find the remaining angle \( A \) using the triangle angle sum property, which states that the sum of angles in a triangle is 180 degrees:

\[A = 180^\circ - B - C_1 = 180^\circ - 29^\circ - 14^\circ = 137^\circ\]

With angles \( A = 137^\circ \), \( B = 29^\circ \), and \( C = 14^\circ \) established, we can now find the remaining side \( a \) using the Law of Sines again:

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

Rearranging gives us:

\[a = \frac{b \cdot \sin A}{\sin B}\]

Substituting the known values:

\[a = \frac{4 \cdot \sin(137^\circ)}{\sin(29^\circ)}\]

Calculating this yields approximately \( 5.63 \). This result is consistent with the triangle's properties, as it is the longest side opposite the largest angle.

In summary, we have determined that the triangle has one valid solution with angles \( A = 137^\circ \), \( B = 29^\circ \), \( C = 14^\circ \), and side lengths \( b = 4 \), \( c = 2 \), and \( a \approx 5.63 \).

Here’s what students ask on this topic:

What is the Law of Sines and how is it used to solve triangles?

The Law of Sines is a fundamental equation in trigonometry used to solve non-right triangles. It states that the ratio of the sine of an angle to its opposite side is constant for all three angles in a triangle. The formula is:

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

To use the Law of Sines, you need to know at least one angle-side pair and one additional angle or side. You can then set up a proportion to solve for the unknown side or angle. This method is particularly useful for solving ASA, SAA, and SSA triangles.

How do you solve an SSA triangle using the Law of Sines?

Solving an SSA triangle with the Law of Sines can be tricky because it may yield one, two, or no solutions. Here’s a step-by-step process:

1. Set up the Law of Sines equation: $\frac{\sin \left(A\right)}{a}=\frac{\sin \left(B\right)}{b}$
2. Solve for $\sin \left(B\right)$: $\sin \left(B\right)=\frac{b\cdot \sin \left(A\right)}{a}$
3. Check if the value of $\sin \left(B\right)$ is between -1 and 1. If not, there is no solution.
4. Find angle B using the inverse sine function: $B=\sin ⁻¹\left(\sin \right(B\left)\right)$
5. Calculate the possible second angle: $B=180-B$
6. Check if the sum of the known angle and each possible B is less than 180°. If so, both solutions are valid.
7. Use the Law of Sines again to find the remaining sides and angles.

What are the different types of triangles where the Law of Sines can be applied?

The Law of Sines can be applied to three types of triangles:

1. ASA (Angle-Side-Angle): Two angles and the included side are known.
2. SAA (Side-Angle-Angle): Two angles and a non-included side are known.
3. SSA (Side-Side-Angle): Two sides and a non-included angle are known. This is also known as the ambiguous case because it can yield one, two, or no solutions.

In each case, the Law of Sines helps to find the missing sides and angles by setting up proportions based on the known values.

How do you solve for a missing side using the Law of Sines?

To solve for a missing side using the Law of Sines, follow these steps:

1. Identify the known angle-side pairs and the angle opposite the missing side.
2. Set up the Law of Sines equation: $\frac{\sin \left(A\right)}{a}=\frac{\sin \left(B\right)}{b}=\frac{\sin \left(C\right)}{c}$
3. Choose the ratio that includes the missing side and the known values.
4. Rearrange the equation to solve for the missing side: $a=\frac{\sin \left(A\right)\cdot c}{\sin }$
5. Plug in the known values and calculate the missing side.

This method allows you to find the length of a side when you know two angles and one side or two sides and one angle.

What is the ambiguous case in the Law of Sines?

The ambiguous case in the Law of Sines occurs when solving SSA (Side-Side-Angle) triangles. In this scenario, you have two sides and a non-included angle, which can lead to three possible outcomes:

1. No Solution: If the calculated sine value is greater than 1, no triangle can be formed.
2. One Solution: If the calculated angle and the given angle sum to less than 180°, only one triangle is possible.
3. Two Solutions: If the calculated angle and the given angle sum to less than 180°, and the supplementary angle also forms a valid triangle, two triangles are possible.

This ambiguity arises because the sine function can yield two different angles with the same sine value, leading to multiple possible triangle configurations.