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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here is a bundle of recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Consider the triangle with vertices A, B, and C.
The applet below shows two different cases for a triangle with vertices A, B, and C. Case I shows the lengths of two sides and the measure of their included angle. Case II shows the lengths of all three sides.
When the Law of Sines cannot be used to solve triangles, the Law of Cosines may be applied.
Consider △ABC with sides of length a, b, and c, which are respectively opposite the angles with measures A, B, and C.
The following equations hold true with regard to △ABC.
a2=b2+c2−2bccos(A)
b2=a2+c2−2accos(B)
c2=a2+b2−2abcos(C)
x2+h2=c2
Commutative Property of Addition
The altitude of the triangle is the perpendicular segment from B to the extension of the base AC. Let D be the endpoint of this segment and x be the distance from D to A.
(a+b)2=a2+2ab+b2
Commutative Property of Addition
In a triangle, when the lengths of two sides and the measure of their included angle are known, the missing side length can be found by applying the Law of Cosines.
Kriz wants to determine the distance between two trees on the other side of the river. Kriz uses a tool that measures the distances to objects. The tool is able to find the distances to each tree as 4 meters and 3.5 meters.
The angle between these sides, from where the Kriz stands with the measuring tool, measures 67∘. Find the distance between the trees, and round the answer to the nearest tenth of a meter.
Start by naming the vertices and sides of the triangle.
For simplicity, the vertices and sides of the triangle will be named.
When all the three side lengths of a triangle are known, the Law of Cosines can be used to find the measure of the angles.
Ramsha lives near a lighthouse. As she likes to observe the landscape, she notices that the light rays coming out of the lighthouse create an angle. She decides to ask the lighthouse keeper, but he insists on not telling her the measure of the angle. She sees a blueprint on the desk behind him, and quickly writes the lengths shown in the diagram before the grumpy keeper blocks her view!To find m∠L, use the Law of Cosines. Then, the measures of other two angles can be found by using either the Law of Cosines or the Law of Sines.
As the diagram indicates, the light rays form a triangle. In this triangle, the three side lengths are known.
The measure of the three angles will be found one at a time.
Substitute values
Calculate power
Multiply
Add terms
LHS−27925=RHS−27925
LHS/(-27300)=RHS/(-27300)
-b-a=ba
Rearrange equation
ba=b/300a/300
cos-1(LHS)=cos-1(RHS)
LHS⋅105=RHS⋅105
Rearrange equation
sin-1(LHS)=sin-1(RHS)
Use a calculator
Round to nearest integer
In △ABC, all three side lengths and the measure of the angle at C are given. Examine how the length of AB changes as the measure of ∠C varies.
m∠C | c2=a2+b2−2abcosC | Relationship between a2+b2 and c2 | Conclusion |
---|---|---|---|
m∠C<90∘ | c2=a2+b2−2ab>0cosC | c2<a2+b2 | If ∠C is acute, not too many conclusions can be made. The opposite side to ∠C can be the largest side, the shortest side, or none. |
m∠C=90∘ | c2=a2+b2−2ab=0cos90∘ | c2=a2+b2 | The cosine of 90∘ is 0. In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to ∠C is the largest side of the triangle. |
90∘<m∠C<180∘ | c2=a2+b2−2ab<0cosC | c2>a2+b2 | If ∠C is obtuse, its measure is greater than the measures of ∠A and ∠C. Therefore, its opposite side is the largest side of the triangle. |
Use the speed formula to calculate the distance traveled before the change in direction SD, and the distance traveled after the change DN.
First, SD and DN will be found. Then, the Law of Cosines will be used to find NS, the distance between New York and San Juan.
Speed=TimeDistance | ||
---|---|---|
Length | SD | DN |
Substitution | 330=3SD | 330=2DN |
Calculation | SD=990mi | DN=660mi |
In this course, the use of the Law of Cosines in solving any type of triangle has been studied. By using this law, the challenge presented at the beginning can be solved.
Zosia knows the lengths of two sides of a triangle and the measure of their included angle. Let A, V, and D denote the vertices of the Summer Triangle.
The Law of Cosines states the relationship between the side lengths of a triangle and the cosine of one of the angles.
Triangle ABC has a perimeter of (11+31) inches. What is the length of b if m∠C=60∘ and a<b.
We have been told that m∠ C=60^(∘). Let's add this value to the diagram.
Using the Law of Cosine, we can write the following equation. (sqrt(31))^2=a^2+b^2-2abcos 60^(∘) Since we have two sides in the triangle that are unknown, we need to use a second relationship in order to determine what they are. Well, we know that the triangle's perimeter is (11+sqrt(31)) centimeters. This provides us with that second expression. a+b+sqrt(31)=11+sqrt(31) ⇓ a= 11-b Now we will substitute 11-b for a into our first equation and simplify.
This is a quadratic equation which we can solve by using the Quadratic Formula.
Since there are two values, b could be 5 or 6 inches. If b is 5 inches, the length of a would consequently be 6 inches. However, we know that a
The cube measures x centimeters on each side. Each point A and B coincides with one of the cube's vertices, and C is the midpoint of a side of the cube. What is the measure of ∠v? Round the measure to the nearest whole degree.
Let's draw AC in a such a way that △ ABC is created where ∠ v is one of its angles. We can label the sides of this triangle as a, b, and c.
Let's determine the lengths of these sides one length at a time.
We can see that side a is the hypotenuse of a right triangle where the legs, x and x, are sides of the cube.
Let's calculate an expression for a using the Pythagorean Theorem. Take care to substitute the correct variables into the theorem according to the legs and hypotenuse.
Side b is also the hypotenuse of a right triangle.
Therefore, we can determine an expression for b with the Pythagorean Theorem as well. Again, take care to substitute the correct variables into the theorem according to the legs and hypotenuse.
The third side is the hypotenuse of a right triangle as well. To identify its length, we will need to draw a diagonal d at the bottom of the cube.
Notice that we already know the length of d as we determined the length of a diagonal of one of the cube's sides when we determined the length of the side we labeled a. We then found that the length of the diagonal is xsqrt(2).
Let's calculate c using the Pythagorean Theorem.
We now have each measurements necessary to use the Law of Cosine in our efforts to determine m∠ v.
Here we will take the inverse of cosine to solve for v.
As we can see m∠ v is about 72^(∘).