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13. Solving Triangles Using the Law of Sines
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Chapter 2
13. 

Solving Triangles Using the Law of Sines

Grasping how to solve triangles with the law of sines is essential for numerous real-world applications. This mathematical principle shines when we possess a triangle with a known angle and side length, or when two angles are known without the side between them. By employing the law of sines, we can unveil the unknown sides or angles. This becomes invaluable in professions like architecture, engineering, and navigation. Whether you're determining a building's height with a given angle and distance or setting a sailing course, the law of sines proves to be a crucial asset.
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Lesson Settings & Tools
11 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Solving Triangles Using the Law of Sines
Slide of 11
In this lesson, a new relation between side lengths and angle measures of a triangle will be derived and used.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

Find the value of If needed, write the answer to the nearest tenth.

a
triangle
b
right triangle
c
right triangle

Calculate the area of the following triangles. If needed, write the answer correct to the nearest integer.

d
An obtuse triangle with a base of 3 and a height of 12. The obtuse angle is at the bottom-left vertex.
e
acute triangle
Challenge

Investigate Triangles With Two Sides and an Angle Known

Can the measure of be found with the given information? If so, what is its value?

Triangle ABC has sides AC and BC measuring 15 and 20 units respectively. The angle A of the triangle measures 70 degrees.
Explore

Relationship Between a Triangle's Angles and Their Opposite Sides

In all the side lengths and angle measures are given. Calculate the ratio of the sine of each angle to the length of its opposite side.
triangle
What conclusions can be made?
Discussion

Law of Sines

One conclusion obtained in the previous exploration can be generalized to all triangles.

For any triangle, the ratio of the sine of an angle to its opposite side is constant.
Triangle with one interactive vertex
Based on the characteristics of the diagram, the following relation can be proven true.

An alternative way to write the Law of Sines is involving the ratio of a side length to the sine of its opposite angle.

Proof

Acute Triangles

Consider an acute triangle with height drawn from one of its vertices. Because is perpendicular to the base, the original triangle is split into two right triangles.

Triangle split into two right triangles
In these two right triangles, and are both opposite angles to Therefore, by applying the definition of the sine ratio to and it is possible to relate the sine of these angles with the side lengths and
Showing sine of angles A and B
Next, can be isolated and written in terms of the corresponding side length and angle, for both right triangles.
By the Transitive Property of Equality, it can be stated that and are equal.
Finally, the obtained equation can be rearranged to obtain the desired result.
▼
Simplify
DivEqn

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

By following the same procedure but drawing the height from vertex it can be shown that Putting these two results together, the Law of Sines is obtained.

Obtuse Triangles

An obtuse triangle will now be considered.

Obtuse Triangle

This proof is very similar to the proof for acute triangles, but it uses an interior and an exterior height. First, the height from the vertex where the obtuse angle is located will be drawn. Just as before, this generates two right triangles.

Drawing inner altitude for an obtuse triangle
The sine ratio will be written for these right triangles.
Showing sine of angles A and C
In the equations, can be isolated.
By the Transitive Property of Equality, it can be stated that and are equal.
The obtained equation can be rearranged to obtain the desired result.
▼
Simplify
DivEqn

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

Next, the exterior height from vertex will be drawn. Let be the point of intersection of this height and the extension of
Obtuse Triangle Exterior Height
Here, and form a linear pair and are therefore supplementary angles. Because the sine of supplementary angles is the same, the sine of equals the sine of Also, using the sine ratio on it can be stated that the sine of is the ratio of to
By the Transitive Property of Equality, the sine of can be written in terms of and
Now will be considered. By using the sine ratio, it follows that the sine of is the ratio of to
Obtuse Triangle
Now, can be written in terms of and and in terms of and
The Transitive Property of Equality can be used one more time.
If only is considered, then can be named as
Obtuse Triangle
This allows to be written as Therefore, and this equation can be rearranged to obtain the desired formula.
▼
Simplify
DivEqn

CancelCommonFac

Cancel out common factors

SimpQuot

Simplify quotient

Finally, this result can be combined with the previous determination to derive the Law of Sines.

The Law of Sines can be used to find side lengths and angle measures of any triangle.
Example

Using the Law of Sines To Find the Side of a Triangle

As previously stated, the Law of Sines can be used to find side lengths of a triangle.

Magdalena will go golfing after school. The locations of her school, the golf course, and her house form a triangular shape. She knows the measures of two of the triangle's angles, and she knows the distance from her house to the school is kilometers.
triangle
What is the distance between the golf course and Magdalena's house? Write the answer rounded to three significant figures.

Hint

Start by labeling the sides and the angles of the triangle.

Solution

The sides of the triangle will be labeled and Similarly, the angles will be labeled and

triangle
By the Law of Sines, the ratio of a side length to the sine of its opposite angle is constant. With this information, a proportion that relates and can be written.
The values and can be substituted into this formula. Then the resulting equation can be solved for the distance between Magdalena's house and the golf course.
SubstituteValues

Substitute values

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

RoundSigDig

Round to significant digit(s)

It was found that the distance between the golf course and Magdalena's house is kilometers, rounded to three significant figures.
Pop Quiz

Practice Finding Sides Using the Law of Sines

In the following applet, represents the side length of a triangle. By using the Law of Sines and, if needed, the Triangle Angle Sum Theorem, find the value of Write the answer rounded to two decimal places.

triangles
Example

Using the Law of Sines To Find Angles of a Triangle

The Law of Sines can also be used to find angle measures of a triangle.

Emily will go backpacking across South America! She will visit Buenos Aires, Santiago, and Asunción, among other cities. Emily knows that the distances from Buenos Aires to Santiago and Asunción are and kilometers, respectively. She also knows that the angle whose vertex is located at Santiago and whose sides pass through Buenos Aires and Asunción measures
triangle
To optimize her journey, she wants to find the measure of the remaining two angles of the triangle formed by these three cities, as well as the distance between Santiago and Asunción . Help Emily find them! Write the answers to the nearest integer.

Hint

Start by labeling the sides and the angles of the triangle.

Solution
The angles of the triangle will be labeled and Similarly, the sides will be labeled and
triangle
By the Law of Sines, the ratio of the sine of an angle to the length of its opposite side is constant. With this information, a proportion that relates and can be written.
In the above formula, and can be substituted. Then the resulting equation can be solved for the measure of the angle formed at Asunción.
SubstituteValues

Substitute values

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Rounded to the nearest degree, it was found that the angle formed at Asunción measures
triangle
Finally, to find the measure of the angle at Buenos Aires, the Triangle Angle Sum Theorem can be used.
This information can be added to the diagram.
triangle
Finally, the Law of Sines can be used again to find the distance between Santiago and Asunción. To do so, recall that the ratio of the length of a side to the sine of its opposite angle is constant.
Next, and will be substituted into the above equation.
SubstituteValues

Substitute values

▼
Solve for
MultEqn

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Emily has all she needs to have a great journey!
Pop Quiz

Practice Finding Angles Using the Law of Sines

In the following applet, represents the measure of an angle of a triangle. By using the Law of Sines and, if needed, the Triangle Angle Sum Theorem, find the value of Write the answer as a single number rounded to the nearest degree, without the degree symbol.

Example

Using the Law of Sines To Find Measurements of a Triangle

The Law of Sines can be used to find missing side lengths and angle measures of a triangle. When those side lengths and angle measures are known, the area and perimeter of a triangle can also be found.

Emily is planning to continue her travels. This time, she wants to visit some cities in the UK, as well as Ireland. She would really like to visit London, Edinburgh, and Dublin. It becomes clear to her that these three cities form a triangle. She is able to figure out that the angles at London and Edinburgh measure and respectively. She also knows that the distance between these two cities is kilometers.
triangle
To optimize her journey, Emily wants to find the perimeter and area of the triangle. Help her with her plan by calculating the perimeter and area! Write the perimeter rounded to the nearest integer and the area rounded to the nearest thousand.

Hint

The perimeter of a triangle is the sum of its three side lengths. The area of the triangle can be found by calculating half the product of two side lengths and the sine of their included angle.

Solution

The perimeter and the area of the triangle will be calculated one at a time.

Perimeter

The perimeter of a triangle is the sum of its three side lengths. Therefore, the missing side lengths need to be found. To make things clearer, the angles and sides will first be labeled.
triangle
Note that the missing angle is opposite to the side of the triangle whose length is known. Therefore, in order to use the Law of Sines to find the missing side lengths, the angle at Dublin must be calculated. To do this, the Triangle Angle Sum Theorem can be used.
The measure of the angle at Dublin is
triangle
The Law of Sines can now be used to find the missing side lengths. The law states that, for every triangle, the ratio of the length of a side to the sine of its opposite angle is constant. With this rule in mind, a proportion involving and can be written.
Next, and can be substituted in the above equation, and can be isolated.
SubstituteValues

Substitute values

▼
Solve for
MultEqn

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The distance between Dublin and Edinburgh is about kilometers. By following the same procedure, the distance between London and Dublin can be found.
Law of Sines Substitute Simplify
The distance between London and Dublin is about kilometers.
triangle
The perimeter of the triangle formed by the three cities can now be calculated.

Area

The area of a triangle can be found by calculating half the product of two side lengths and the sine of their included angle. Since all angle measures and sides lengths are known, any of them can be used. In this case, and will be arbitrarily used.
The respective values can be substituted to find the area of the triangle.
SubstituteValues

Substitute values

▼
Evaluate right-hand side
Multiply

Multiply

MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

Approximate to the nearest thousand

The area of the triangle formed by London, Dublin, and Edinburgh is about square kilometers. Thanks for helping Emily plan her travels!
Discussion

The Ambiguous Case of the Law of Sines

The Law of Sines is useful to find missing angle measures and side lengths for any type of triangle.
triangle
Consider a triangle with vertices and where the measure of the angle at is and the lengths of and are and centimeters, respectively. The following diagram represents the triangle that was just described.
triangle
The measure of the angle at can be found by using the Law of Sines.
▼
Solve for
MultEqn

CommutativePropMult

Commutative Property of Multiplication

MoveLeftFacToNum

ReduceFrac

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Using the Law of Sines, it was found that the measure of the angle at is about
triangle
Is this the only possible measure for the angle at To find the measure of this angle, the Law of Sines was applied. That is, for any triangle, the ratio of the sine of an angle to the length of its opposite side is constant.
Recall now that the sine of an angle is equal to the sine of the supplement of the angle.
With this information, it can be stated that the sine of an angle whose measure is is equal to the sine of an angle whose measure is Therefore, by the Substitution Property of Equality, a new proportion can be derived.
The obtained proportion implies a new triangle that satisfies the initial conditions.
obtuse triangle
Therefore, in the given context, there are two valid measures for the angle at which are and Consequently, the measure of a missing angle is ambiguous. This shows a limitation of the Law of Sines. To avoid this ambiguity, it must be said whether the missing angle is acute or obtuse, or a diagram of the triangle should be provided.
Closure

Calculating an Angle of a Triangle Using the Law of Sines

The challenge presented at the beginning of this lesson can be solved by using the Law of Sines.

triangle
Find the measure of the angle at Write the answer rounded to the nearest degree.

Hint

In a triangle, the ratio of the sine of an angle to the length of its opposite side is constant.

Solution
Since the ratio of the sine of an angle to the length of its opposite side is constant, a proportion can be written.
The above equation can be solved for
▼
Solve for
MultEqn

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The measure of the angle at is about
triangle


Level 1
Level 2
Level 3
Solving Triangles Using the Law of Sines
Exercises

Ignacio has bought a very large American flag — it includes a support beam — that he wants to raise outside his house. The diagram gives the dimensions of the flag beam in meters.

an American flag hanging of a wall
 What is the greatest distance from that Ignacio can place the end of the support beam if he wants to be Round the answer to two decimal places.
What is the shortest distance from that we can place for be Round the answer to two decimal places.

Let's draw a possible triangle that can be created by the flag, the support beam, and the wall. We will label the point where the support beam meets the flagpole as B.

We can use the Law of Sines to determine the length of PA. To first determine that length, we need to know the measure of m∠ B. For that angle measure, we need to know the measure of m∠ A. Then we will have enough information to use the Law of Sines. Now, let's solve for m∠ A.

Now we can use the Interior Angles Theorem to determine the unknown angle. m∠ B+30^(∘)+36.2215...^(∘) =180^(∘) ⇓ m∠ B=113.7784...^(∘) With this information, we can use the Law of Sines to calculate the length of PA.

As we can see, when PA equals 6.04 meters, the angle at P is 30^(∘).


Notice that sin θ=sin(180^(∘)-θ). Therefore, ∠ A can have a second measure. m∠ A = 180^(∘)-36.2215...^(∘) = 143.7784...^(∘) Since AB

Now we can use the Law of Sines to calculate this second possible length of PA.

As we can see, PA could also have a length of 0.72 meters.

For a school science project, Ali and Bjorn were given devices called sextants, which can be used for precisely measuring angles of elevation between specific locations on Earth and a point in the night sky. One night, Ali sees the Aurora borealis, known as the northern lights, from his house. His friend Bjorn, who lives kilometers away, also sees the lights. Ali and Bjorn want to know how high above the ground the lights are. Therefore, they measure the angles of elevation from their respective houses to a specific point in the sky.

picture of the northern lights in Sweden
 What is the vertical distance to the northern lights if and Round the answer to the nearest kilometer.
Construct a formula that will allow Ali and Bjorn to measure the height of the northern lights for any given angles and Simplify the formula as much as possible.

In the figure, we can identify two right triangles inside of the greater triangle ABC. Notice that we know the measures of two angles of △ ABC which means we can find the measure of ∠ C by using the Interior Angles Theorem. 83^(∘)+87^(∘)+m∠ C = 180^(∘) ⇓ m∠ C = 10^(∘) Let's add this finding to the triangle.

With the given information, we can find the length of either AC or BC by using the Law of Sines. Let's find BC.

Let's keep the side length in exact form for precision purposes and add it to the diagram. At the same time, we will focus on the right triangle BDC.

Let's substitute the hypotenuse, the given acute angle, and its opposite side of △ BDC in the sine ratio and solve for h.

The height from the ground to the northern lights is about 171 kilometers.

Instead of numerical values, we have A and B. This means we can express the angle at C as 180^(∘)-(A+B).

Now we can use the Law of Sines to write a general expression to describe the length of BC.

As in Part A, we will use the sine ratio to find an expression for h.


Solving Triangles Using the Law of Sines

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