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Similarity, Proof, and Trigonometry

Application of the Laws of Sine and Cosine

This lesson will explore the applications of the Law of Sines and the Law of Cosines to solve different types of triangles. Furthermore, real-life applications of these ideas will also be explored.

Catch-Up and Review

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

Explore

Which Law Should be Applied?

Take a look at the following triangles. Think whether they can be solved by using the Law of Sines or the Law of Cosines.

Different triangles

Challenge

Area of a Circle Circumscribed Around a Non-Right Triangle

The following figure shows a circle circumscribed around a non-right triangle. Notice that none of the triangle's sides correspond to the diameter of the circle. What is the area of the circle?

Circle Circumscribed Around an Oblique Triangle

Discussion

Solving Triangles Using the Law of Sines and the Law of Cosines

Trigonometric ratios are often used to solve right triangles, but they cannot be used to solve non-right, or oblique, triangles. For these triangles, the Law of Sines and the Law of Cosines are particularly useful because they can be used to solve any triangle, right or oblique. To solve a triangle using the Law of Sines or the Law of Cosines, three pieces of information must be known.

Case Given Information Law Strategy
Two angles and a side length Law of Sines The Triangle Angle Sum Theorem can be used to find the missing angle measure. Then the Law of Sines can be used to find the unknown side lengths.
Two side lengths and a non-included angle Law of Sines The Law of Sines can be used to solve for one of the unknown angle measures. Then the Triangle Angle Sum Theorem can be used to find the third angle measure. Finally, the Law of Sines can be applied one more time to find the unknown side length.
Three side lengths Law of Cosines The Law of Cosines can be used to find any of the unknown angle measures. Then, either the Law of Sines or the Law of Cosines can be used to find another missing angle measure. Finally, the Triangle Angle Sum Theorem can be used to find the third angle measure.
Two side lengths and their included angle Law of Cosines The Law of Cosines can be used to find the missing side length. Then, either the Law of Cosines or the Law of Sines can be used used to find a missing angle measure. Finally, the Triangle Angle Sum Theorem can be used to find the last angle measure.
In the following applet, a triangle and three of its parts are shown. Analyze the given information to decide which law can be used to solve the triangle.

Pop Quiz

Analyzing Different Triangles to Determine Which Law Can Be Used

Different triangles and strategies to solve them

Example

Calculating the Height of the Leaning Tower of Pisa

Zain is vacationing in Italy. They were in Pisa to see the famous Leaning Tower when a question came across their mind. What would the tower's height be if it was not a leaning tower? Zain's distance to the tower is meters and they can measure an angle of elevation of to the tower's top. Furthermore, the guidebook states that the tower's inclination is about

Diagram with the Tower of Pisa

Remembering a Geometry lesson, they realize that the situation can be modeled using a non-right triangle. Help Zain calculate the height of the upright tower. Write the answer rounded to one decimal place.

Hint

The inclination angle and its adjacent angle are complementary angles.

Solution

The situation can be modeled using a non-right triangle.

Diagram with the Tower of Pisa and labels
The tower's inclination angle and are complementary angles, so the sum of their measures adds to The measure of can now be calculated. Now that two angles and their included side are known, it is possible to use the Law of Sines. To do so, the measure of must be found. This will be done by using the Triangle Angle Sum Theorem. Finally, the Law of Sines will be used to write a proportion. This equation will be solved for the height of the upright tower.
Solve for
Therefore, the height of the Leaning Pisa Tower would be around meters if it was standing straight.

Pop Quiz

Measuring Trees Using the Law of Sines

Magdalena is doing some research in the forest for a biology project. She has a device that allows her to measure angles. She can also measure the distance from a tree to her device. To help Magdalena complete her research project, find the lengths of different trees, rounded to one decimal place.

Interactive graph showing different trees and triangles

Discussion

Extending the Definition of Sine and Cosine to Obtuse Angles

As mentioned before, the Law of Sines and the Law of Cosines are valid for all types of triangles, including both right and non-right triangles. However, the definitions of the sine and cosine of an angle are given in terms of the ratios of a right triangle's sides.

Sine and Cosine of an angle defined in terms of a Right triangle's sides

Therefore, these definitions do not seem to be compatible with obtuse angles. Nevertheless, they can be extended to deal with obtuse angles by considering the following identities.

Rule

Sine and Cosine of Supplementary Angles

The sine of supplementary angles are equal. Conversely, the cosine of supplementary angles are opposite values.

The graph shown below allows to verify both identities for different angles.
Graph comparing the sine and cosine of two supplementary angles
These identities can also be used when the angle is given in radians.

As shown, these identities allow the definition of the value of the sine and cosine of obtuse angles and the application the Law of Sines and the Law of Cosines to any triangle.

Example

Calculating the Perimeter of a Piece of Land

Ignacio's grandparent wants to construct a fence for a quadrilateral piece of land. To find the perimeter of the land, he starts measuring its sides using an old trundle wheel. Unfortunately, after measuring just two sides, the trundle wheel breaks.

Quadrilateral piece of land

Ignacio wants to help his grandparent and, in an attempt to simplify the problem, he divides the land into two triangles. Then, by using a compass, he is able to measure the angles of these triangles.

Land divided into two triangles and showing angle measures
Help Ignacio find the perimeter of the piece of land, rounded to nearest integer.

Hint

Since two side lengths and the measure of their included angle are known in the Law of Cosines can be used to solve for the missing side length.

Solution

Note that two side lengths and all the angle measurements are known in

Triangle ABC
In particular, since two side lengths and the measure of their included angle are known, the Law of Cosines can be used to solve for the missing side length.
Solve for
When solving the above equation, only the principal root was considered because side lengths are always positive. Therefore, the length of is about meters.
Quadrilateral divided into two triangles

Now, will be considered.

Triangle ACD
Since one side length and all the angle measures are known, the missing side lengths can be found by using the Law of Sines. The length of will be calculated.
Solve for
Therefore, the length of is about meters. Next, the length of can be calculated by following the same procedure.
Solve for
Now, all the sides of the piece of land are known.
Piece of land with all side lengths shown

Finally, the perimeter will be calculated by adding all the side lengths. The perimeter of the piece of land is about meters.

Pop Quiz

Law of Cosines in Soccer Practice

Kriz is setting up for a free shots on an empty goal. When considering their distance to both goal posts, they realize that the Law of Cosines can be used to calculate the top measure of the angle in which they must kick the ball in order to score. They are practicing with a standard meter net. Help Kriz calculate this angle and score the goal! Write the answer rounded to one decimal place.

triangle with three known sides formed by soccerplayer and the goal net

Example

Locating of a Smartphone Using Triangulation

A burglar robbed a store and took the cashier's smartphone. In an attempt to outsmart the police, the burglar turned off the phone's GPS. Once at their secret location, the burglar felt safe and made a call to plan their next move. However, the smartphone signal was detected by two nearby towers, estimating their distance to the phone in use.
Iteractive diagram showing houses and cellphone towers working
With the distance from the towers to the location of the smartphone and knowing that the towers are blocks apart, the police think they have enough information to locate the burglar. Help the police find the burglar's location! Write the horizontal and vertical distance in blocks with respect to the phone tower at the left. Round the answers to the nearest integer.

Hint

The situation can be modeled using a triangle with three known side lengths.

Solution

The distance from the tower at the left to the smartphone is about blocks. The distance from the tower at the right to the phone is about blocks. The towers are blocks apart. Therefore, the situation can be modeled using a triangle with three known sides lengths.

Simplified diagram of the situation showing a triangle with the side lengths in blocks
Even if no angle is known at first, knowing the three side lengths allows the use of the Law of Cosines to solve for any angle. In this case, it is convenient to solve for which corresponds to the vertex represented by the left tower.
Solve for

Now that the measure of is known, the horizontal and vertical distance from the left tower to the smartphone, and respectively, can be calculated. To do this, a right triangle formed by the left tower and the phone will be considered. Then, trigonometric ratios will be used.
Right triangle formed by the hideout and the left tower
The vertical distance to the smartphone is the opposite leg to the angle whose measure is Therefore, the length of the leg can be found by using the sine ratio.
Solve for
Similarly, the horizontal distance is the adjacent leg to the angle whose measure is Therefore, the cosine ratio can be used to find the value of
Solve for
The horizontal distance from the left tower to the burglar's location is about blocks, and the vertical distance is about blocks.

Example

Finding the Altitude of a Helicopter

Two radar stations located kilometers apart detect a passing helicopter. The first station measures an angle of elevation to the helicopter of while the second station measures an angle of elevation of At what altitude, rounded to one decimal place, is the helicopter flying?
Oblique triangle with a helicopter and two antennas on its vertices

Hint

The missing angle can be found by using the Triangle Angle Sum Theorem.

Solution

Because two angles and the included side are known, this problem can be approached by using the Law of Sines to find the distance between the helicopter and one of the radar stations. Once one of the missing side lengths is determined, trigonometric ratios can be used to find the altitude, or height, of the helicopter. The first thing to do is to find the unknown angle.

Interior angles and sides of the triangle previously introduced
To do this, the Triangle Angle Sum Theorem can be used. The measure of will now be found. Now that the opposite angle to the known side length has been found, the Law of Sines can be used to write a proportion and solve for any of the missing sides. For instance, can be calculated.
Solve for
The distance from Station to the helicopter is about kilometers. This distance is the hypotenuse of the right triangle formed by considering the helicopter and Station as vertices. Furthermore, in this right triangle, the opposite leg to the angle that measures represents the helicopter's altitude.
Right triangle showing the helicopter, station 1, and the height
Finally, the sine ratio can be used to find the altitude of the helicopter.
Solve for
The helicopter is flying at an altitude of about kilometers.

Discussion

Connection Between the Law of Sines and the Circumcircle of a Triangle

The Law of Sines states that for any triangle, the ratio of the sine of an angle to the length of its opposite side is constant. However, this is not just any constant. In fact it has an important geometrical interpretation.

Rule

Extended Form of the Law of Sines

For any triangle, the diameter of its circumcircle is equal to the ratio of a side length to the sine of its opposite angle.

Considering the above figure and letting be the diameter of the circle, the following equation holds true.

Closure

Calculating the Area of a Circle Circumscribed Around a Non-Right Triangle

The challenge presented at the beginning of this lesson can be solved by using a combination of the Law of Sines, the Law of Cosines, and the extended form of the Law of Sines.

Circle Circumscribed Around an Oblique Triangle

The question here was to find the area of the circle. This will be answered by first finding the ratio of the triangle's side lengths to the sine of their opposite angles.

a What is the ratio of a side length to the sine of its opposite angle? Write the answer rounded to one decimal place.
b What is the area of the circumscribed circle? Write the answer rounded to one decimal place.

Hint

a Two side lengths and the measure of the included angle are known.
b The area of a circle is where is the radius.

Solution

Since two side lengths and the included angle are known, the Law of Cosines can be used to find the missing side length. Because the missing side is opposite to the known angle, finding the side length will allow to calculate the desired ratio.

Circle Circumscribed Around an Oblique Triangle
The Law of Cosines will be used to find the missing side
Solve for
When solving the above equation, only the principal root was considered. This is because a side length is always positive. Therefore, the missing side length is about units.
Circle Circumscribed Around an Oblique Triangle
Now that the measure of and the length of its opposite side are known, the desired ratio can be calculated.
Evaluate
The ratio of a side length to the sine of its opposite angle is about Therefore, by the extended form of the Law of Sines, the length of the circle's diameter is about Recall that the radius of a circle is half its diameter. With this information, the area of the circle can be calculated.
Simplify right-hand side
The area of the circumscribed circle is about square units.
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