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Similarity, Proof, and Trigonometry

Solving Triangles Using the Law of Sines

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
obtuse triangle
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

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
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
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
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.
Solve for
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.
Solve for

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.
Solve for
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.
Solve for
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.
Evaluate right-hand side

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

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

The measure of the angle at is about
triangle
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