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| 14 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a some recommended readings before getting started with this lesson.
Take a look at the following triangles. Think whether they can be solved by using the Law of Sines or the Law of Cosines.
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?
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 |
---|---|---|---|
1 | 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. |
2 | 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. |
3 | 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. |
4 | 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. |
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 80 meters and they can measure an angle of elevation of 37∘ to the tower's top. Furthermore, the guidebook states that the tower's inclination is about 4∘.
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.
The inclination angle and its adjacent angle are complementary angles.
The situation can be modeled using a non-right triangle.
Substitute values
LHS⋅sin37∘=RHS⋅sin37∘
Use a calculator
Rearrange equation
Round to 1 decimal place(s)
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.
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.
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.
The sine of supplementary angles are equal. Conversely, the cosine of supplementary angles are opposite values.
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.
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.
Since two side lengths and the measure of their included angle are known in △ABC, the Law of Cosines can be used to solve for the missing side length.
Note that two side lengths and all the angle measurements are known in △ABC.
Now, △ACD will be considered.
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 7.3 meter net. Help Kriz calculate this angle and score the goal! Write the answer rounded to one decimal place.
The situation can be modeled using a triangle with three known side lengths.
The distance from the tower at the left to the smartphone is about 4.5 blocks. The distance from the tower at the right to the phone is about 2.8 blocks. The towers are 6 blocks apart. Therefore, the situation can be modeled using a triangle with three known sides lengths.
Substitute values
Calculate power
Add terms
Multiply
LHS−56.25=RHS−56.25
LHS/(-54)=RHS/(-54)
Rearrange equation
cos-1(LHS)=cos-1(RHS)
Use a calculator
Round to nearest integer
Substitute values
LHS⋅4.5=RHS⋅4.5
Use a calculator
Round to nearest integer
Rearrange equation
Substitute values
LHS⋅4.5=RHS⋅4.5
Use a calculator
Round to nearest integer
Rearrange equation
The missing angle can be found by using the Triangle Angle Sum Theorem.
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.
Substitute values
LHS⋅sin45∘=RHS⋅sin45∘
Use a calculator
Rearrange equation
Round to 1 decimal place(s)
Substitute values
LHS⋅21.3=RHS⋅21.3
Use a calculator
Rearrange equation
Round to 1 decimal place(s)
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.
The following goes for any triangle. The diameter of a triangle's circumcircle is equal to the ratio of a side length to the sine of its opposite angle.
Now, consider the above figure and let D be the diameter of the circle. With the given information, the following equation holds true.
sinAa=sinBb=sinCc=D
The proof can be completed in two parts. Part I shows how the sides of a triangle are proportional to the sines of the opposite angles. Part II shows this proportion is equal to the diameter of the circle circumscribed around the triangle.
Consider a triangle ABC with side lengths a, b, and c, and angle measures A, B, and C.
Draw the circumcircle of the triangle ABC with its circumcenter O.
By the Inscribed Angle Theorem, the measure of A is half of the measure of the intercepted arc BC.
Now, draw the central angle with the intercepted arc BC. Recall that a central angle and its intercepted arc have the same measure. Therefore, the measure of BC is equal to the measure of m∠COB.
Note that OB and OC are both equal to the radius R. Hence, △COB is an isosceles triangle with two congruent sides OB and OC.
Now, focus on △COB. Draw its altitude from the vertex angle O. Since △COB is an isosceles triangle, the altitude bisects both the vertex angle and the opposite side.
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.
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.
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.
r=2.6
Calculate power
Commutative Property of Multiplication
Use a calculator
Round to 1 decimal place(s)
A group of archaeologists found a pyramid during their exploration trip. They measured the angle of elevation to the apex of the pyramid to be 21∘. They kept walking and after reaching the pyramid, they measured its slope to be 47∘. If they were 230 meters away when they measured the elevation angle, what is the height h of the pyramid? Round the answer to the nearest integer.
Let's start by drawing a triangle to model the situation.
Notice that if m∠ C were known, it would be possible to use the Law of Sines to solve for the slant height of the pyramid a. Nevertheless, the pyramid's slope angle is supplementary to ∠ B. Therefore, the sum of their measures is 180 ^(∘). We can solve for m∠ B. 47 ^(∘) + m∠ B = 180 ^(∘) ⇕ m∠ B = 133 ^(∘) Now m∠ C can be found by using the Interior Angles Theorem. 21 ^(∘) + 133 ^(∘) + m∠ C = 180 ^(∘) ⇕ m∠ C = 26 ^(∘) With the new information at hand and using the Law of Sines, we can set up a proportion to solve for the slant height of the pyramid a. Finding the slant height is important because then we can use trigonometric ratios to find the pyramid's height.
Therefore, the slant height of the pyramid is about 188 meters long. Finally, a cross-section of the pyramid can be modeled using a right triangle.
Because the hypotenuse — slant height — is known and because the height of the pyramid is the opposite leg to the known angle measure 47 ^(∘), the sine definition can be used to solve for the height.
The height of the pyramid is about 137 meters.
Heichi won a contest with his school project and was awarded a cruise ticket. During the trip, the cruise left the port of Nassau in the Bahamas and traveled for about 4 hours. Then, it changed direction by turning 36∘ and arrived in Miami after 6 more hours.
To find the distance from Nassau to Miami, we need to know the distance of NC and CM. We can then use this information to determine the distance between Nassau and Miami.
Distances can be calculated by using the speed formula. The average speed is the distance traveled divided by the amount of time spent traveling. speed = distance/time We know that the ship took 4 hours to travel between Nassau and the point where the it changed direction. The cruise ship had an average speed of 20 miles per hour. Let's substitute these values into the formula and solve for NC.
The distance covered in the first 4 hours of the trip is 80 miles. The distance covered in the next 6 hours, CM, can be calculated in a similar way.
Note that the angle of the turn and its adjacent angle, ∠ C, are supplementary angles. Therefore, the sum of their measures is 180 ^(∘).
With this information we can find ∠ C. 36 ^(∘) + m∠ C = 180 ^(∘) ⇒ m∠ C = 144 ^(∘) With the known information, the situation can be modeled by using a triangle for which the lengths of two sides and the measure of their included angle are known.
Therefore, the Law of Cosines can be used to solve for the missing side length c that represents the distance from Nassau to Miami.
Since a length cannot be negative, only the principal root is considered here. Therefore, the distance from Nassau to Miami is about 191 miles.