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

A S A , a = 5.38 , b = 1.61 , C = 105 ° ASA,a=5.38,b=1.61,C=105\degree A S A , a = 5.38 , b = 1.61 , C = 105°

S A A , a = 6.69 , b = 22.4 , C = 105 ° SAA,a=6.69,b=22.4,C=105\degree S AA , a = 6.69 , b = 22.4 , C = 105°

A S A , a = 6.69 , b = 22.4 , C = 105 ° ASA,a=6.69,b=22.4,C=105\degree A S A , a = 6.69 , b = 22.4 , C = 105°

S A A , a = 5.38 , b = 1.61 , C = 105 ° SAA,a=5.38,b=1.61,C=105\degree S AA , a = 5.38 , b = 1.61 , C = 105°

7. Non-Right Triangles

Law of Sines

7. Non-Right Triangles

Law of Sines: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

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: sinAa = sinBb = sinCc. This method allows for solving angles and sides in triangles classified as ASA, SAA, or SSA, with SSA presenting unique challenges, potentially yielding zero, one, or two solutions. Understanding these classifications is crucial for accurate problem-solving.

concept

Intro to Law of Sines

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Intro to Law of Sines Video Summary

In solving triangles, understanding the distinction between right triangles and non-right triangles is crucial. For right triangles, the Pythagorean theorem and trigonometric ratios, often remembered by the acronym SOHCAHTOA, can be effectively utilized. For instance, if you have a right triangle with a known hypotenuse and an angle, you can easily find the lengths of the other sides using these methods.

However, when dealing with non-right triangles, the approach changes significantly. In such cases, the Law of Sines becomes the primary tool for finding missing sides or angles. The Law of Sines states that the ratio of the sine of an angle to the length of the opposite side is constant across all three angles and sides of the 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. The notation is important: angles are denoted by uppercase letters, while the corresponding sides are represented by lowercase letters.

To apply the Law of Sines, you need to identify two ratios where you know three out of the four variables. For example, if you know angles $A$ and $C$, along with side $c$, you can set up the equation:

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

From here, you can rearrange the equation to solve for 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 instance, if $c = 6$, $C = 70^\circ$, and $A = 30^\circ$, the calculation would yield:

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

Upon performing the calculations, you would find that $a \approx 3.19$. This result aligns with the geometric intuition that in a non-right triangle, the side lengths can differ from those in a right triangle due to the angles involved.

It's also worth noting that in some contexts, you may encounter Greek letters such as alpha, beta, and gamma in place of $A$, $B$, and $C$, but the principles remain the same. Mastering the Law of Sines is essential for solving non-right triangles effectively.

Problem

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

8.20

4.39

2.20

1.61

Problem

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

48.6°

77.2°

40.5°

35.3°

concept

Solving SAA & ASA Triangles

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Solving SAA & ASA Triangles Video Summary

To solve for missing sides and angles in non-right triangles, the law of sines is a crucial 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): Two angles and the side between them are known.
Side-Angle-Angle (SAA): Two angles and a side that is adjacent to them are known.
Side-Side-Angle (SSA): Two sides and an angle that is not between them are known.

In this context, we will focus on the ASA and SAA types. To classify a triangle, sketching it based on the given angles and sides can be very helpful. For example, if you know that angle A is 30 degrees, angle C is 70 degrees, and side c is 6, you can visualize the triangle and confirm that it is an SAA triangle.

Once classified, the next step is to find the third angle using the angle sum formula, which states that the sum of the angles in a triangle equals 180 degrees. Thus, if you have angles A and C, you can find angle B as follows:

\[A + B + C = 180^\circ \implies B = 180^\circ - A - C\]

Substituting the known values gives:

\[B = 180^\circ - 30^\circ - 70^\circ = 80^\circ\]

Now that all angles are known, the next step is to find the missing sides using the law of sines, which states:

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

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

\[\frac{a}{\sin A} = \frac{c}{\sin C}\]

Substituting the known values:

\[a = \frac{c \cdot \sin A}{\sin C} = \frac{6 \cdot \sin(30^\circ)}{\sin(70^\circ)}\]

Calculating this gives:

\[a \approx 3.19\]

Next, to find side b, you can use a similar approach:

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

Substituting the known values:

\[b = \frac{c \cdot \sin B}{\sin C} = \frac{6 \cdot \sin(80^\circ)}{\sin(70^\circ)}\]

Calculating this results in:

\[b \approx 6.29\]

In summary, by 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 a triangle. This systematic approach is essential for mastering triangle problems in trigonometry.

Problem

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

$SAA,a=6.69,b=22.4,C=105\degree$

$ASA,a=6.69,b=22.4,C=105\degree$

$ASA,a=5.38,b=1.61,C=105\degree$

$SAA,a=5.38,b=1.61,C=105\degree$

Problem

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.

$459.6ft$

$195.8 ft$

$152.3ft$

$233.4ft$

concept

Solving SSA Triangles ("Ambiguous" Case)

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Solving SSA Triangles ("Ambiguous" Case) Video Summary

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 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 you a value for $\sin B$. If this value exceeds 1, there is no solution, as the sine function only ranges from -1 to 1. If the value is less than or equal to 1, you can proceed to find angle $B$ using the inverse sine function:

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

This calculation yields two potential angles for $B$: the first angle directly from the inverse sine, and the second angle calculated as:

$$B_2 = 180^\circ - B_1$$

Next, you must check the sum of the angles. If the sum of $A$ and $B_2$ exceeds 180 degrees, then only one triangle is possible. If it is less than 180 degrees, both triangles are valid solutions.

For the first triangle, you can find the third angle $C$ using the angle sum property:

$$C = 180^\circ - (A + B_1)$$

For the second triangle, use:

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

In summary, solving SSA triangles involves careful application of the law of sines, consideration of the possible angles derived from the inverse sine function, and verification of the angle sums to determine the number of valid triangles. This methodical approach allows you to navigate the ambiguity inherent in SSA cases effectively.

example

Example 1

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Example 1 Video Summary

In this example, we explore a triangle with sides $ a = 1 $, $ b = 4 $, and angle $ A = 30^\circ $. This configuration represents an SSA (Side-Side-Angle) triangle, which can lead to ambiguous cases where a triangle may not exist. To analyze this, we can use the Law of Sines, which states:

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

Here, we aim to find angle $ B $. Rearranging the equation gives us:

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

Substituting the known values, we have:

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

Since $ \sin(30^\circ) = \frac{1}{2} $, this simplifies to:

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

However, the sine of an angle cannot exceed 1, indicating that there is no possible angle $ B $ that satisfies this equation. Therefore, we conclude that a triangle cannot be formed with the given dimensions.

To visualize this scenario, consider the angle $ A $ of $ 30^\circ $ and side $ b = 4 $. The side $ a = 1 $ is insufficient to connect to the endpoint of side $ b $, regardless of how it is positioned. This geometric limitation confirms that no triangle can exist under these conditions, leading us to the final conclusion that there is no solution for this triangle configuration.

example

Example 2

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Example 2 Video Summary

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 focus on the known values of $ b $ and $ c $ while disregarding side $ a $ since it is unknown. 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:

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

This calculation yields $ \sin C \approx 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 can yield two possible angles, so we also calculate the second 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. Adding the known angle $ B $ (29 degrees) to $ C_2 $ gives:

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

This sum exceeds 180 degrees, which is not possible for a triangle. Therefore, $ C_2 $ is invalid, leaving us with only one solution: $ C_1 = 14^\circ $.

To find the remaining angle $ A $, we use the triangle angle sum property, which states that the sum of the angles in a triangle is 180 degrees:

\[A + B + C_1 = 180^\circ\]

Substituting the known values:

\[A + 29^\circ + 14^\circ = 180^\circ\]

Solving for $ A $ gives:

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

Now, we have all angles: $ A = 137^\circ $, $ B = 29^\circ $, and $ C = 14^\circ $. To find the remaining side $ a $, we again apply the Law of Sines:

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

Rearranging gives:

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

Substituting the known values:

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

Calculating this results in $ a \approx 5.63 $. This value is consistent with the triangle's properties, as it is the longest side opposite the largest angle.

In summary, the triangle has angles $ A = 137^\circ $, $ B = 29^\circ $, and $ C = 14^\circ $, with sides $ a \approx 5.63 $, $ b = 4 $, and $ c = 2 $. This example illustrates the process of solving for unknown angles and sides in a triangle using the Law of Sines, emphasizing the importance of validating potential solutions within the constraints of triangle geometry.

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Law of Sines

7. Non-Right Triangles

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7. Non-Right Triangles

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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:

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

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 (Angle-Side-Angle), SAA (Side-Angle-Angle), and SSA (Side-Side-Angle) triangles.

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

To solve an ASA (Angle-Side-Angle) triangle using the Law of Sines, follow these steps:

Identify the given angles and the side between them.
Use the angle sum property of triangles to find the third angle: $A + B + C = 180°$ .
Apply the Law of Sines to find the unknown sides. Set up the proportion: $\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$ .
Solve for the unknown sides by cross-multiplying and isolating the variable.

This method ensures you can find all missing sides and angles in the triangle.

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:

ASA (Angle-Side-Angle): Two angles and the included side are known.
SAA (Side-Angle-Angle): Two angles and a non-included side are known.
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 zero, one, or two solutions.

Understanding these classifications helps in setting up the correct proportions and solving for the unknown sides and angles using the Law of Sines.

How do you handle the ambiguous case (SSA) when using the Law of Sines?

The ambiguous case (SSA) occurs when you have two sides and a non-included angle. To handle this:

Set up the Law of Sines proportion: $\frac{sin A}{a} = \frac{sin B}{b}$ .
Solve for the sine of the unknown angle: $sin B = b * sin A / a$ .
Check if the value is between -1 and 1. If not, there is no solution.
If the value is valid, find the angle using the inverse sine function. There may be two possible angles: $B = sin$ ^-1(value) and $B' = 180° - B$ .
Determine if each angle leads to a valid triangle by checking if the sum of angles is less than 180°.

This process helps identify whether there are zero, one, or two possible solutions.

Can the Law of Sines be used to solve right triangles?

While the Law of Sines can technically be used to solve right triangles, it is not the most efficient method. For right triangles, the Pythagorean theorem and trigonometric ratios (SOHCAHTOA) are more straightforward and commonly used. The Law of Sines is particularly useful for solving non-right triangles, where these simpler methods do not apply. In non-right triangles, the Law of Sines helps find unknown sides and angles by setting up proportions based on the sine of the angles and their opposite sides.