Here is a bundle of recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Consider the triangle with vertices and
On a June night, Zosia observes an astronomical asterism, which is called the Summer Triangle.
The applet below shows two different cases for a triangle with vertices and Case I shows the lengths of two sides and the measure of their included angle. Case II shows the lengths of all three sides.
Consider with sides of length and which are respectively opposite the angles with measures and
The following equations hold true with regard to
The first equation will be proven. The other two equations can be proven by following the same procedure. Begin by drawing the altitude from to its opposite side
In the cosine of is the ratio of to Finally, can be substituted for into By doing so, the formula for the Law of Cosines is obtained.
The first equation will be proven for obtuse angles. The remaining equations can be proven similarly. Consider with side lengths of and respectively opposite the angles with measures and such that is greater than
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 meters and meters.
The angle between these sides, from where the Kriz stands with the measuring tool, measures Find the distance between the trees, and round the answer to the nearest tenth of a meter.
For simplicity, the vertices and sides of the triangle will be named.
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.
By the Triangle Angle Sum Theorem, the sum of the interior angles of a triangle is Therefore, all measures of were found.
In all three side lengths and the measure of the angle at are given. Examine how the length of changes as the measure of varies.
|Relationship between and||Conclusion|
|If is acute, not too many conclusions can be made. The opposite side to can be the largest side, the shortest side, or none.|
|The cosine of is In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to is the largest side of the triangle.|
|If is obtuse, its measure is greater than the measures of and Therefore, its opposite side is the largest side of the triangle.|
First, and will be found. Then, the Law of Cosines will be used to find the distance between New York and San Juan.
Approximate to nearest hundred
Let and be the lengths of the sides opposite and respectively. By the Law of Cosines, the following equations holds true for Since the values of and are given, the first equation will give the length of the third side.