Download Law of Cosines

Law of Cosines DISCUSSION
Precalculus
A ( ____________, _____________ )
c
b
C
(0,0)
a
B
(a, 0)
The law of cosines applies when we know two sides of a triangle and the
included angle between the two sides and we want to determine the third side
(side opposite the included angle).
To develop the Law of Cosines, we develop an equation for the length of side c

using the distance formula d 
x2  x1 2   y2  y1 2
 squared:
The Law of Cosines is:
______________________________________
Example 1 Suppose that two sides of a triangle have lengths 3 cm and 7 cm and
that the angle between them measures 1300. Find the length of the
third side.
The Law of Cosines can also be rearranged to solve for an angle. This allows
us to find any angle in a triangle if three side lengths are given. Rearranging,
we obtain:
Example 2 The lengths of the sides of a triangle are 5, 10, and 12. Solve the
triangle. (means find all 3 angle measurements)
θ
12
5
α
β
10
Notice that if cosine of an angle in a triangle is negative, the angle will be an
obtuse angle (angles in the second quadrant have a negative cosine value).
Example 3 A triangle has sides of lengths 6, 12, and 15.
B
12
6
A
7.5
D
7.5
C
a.) Find the measure of the smallest angle.
b.) Find the length of the median to the longest side. (the median is
the segment from a vertex to the midpoint of the opposite side).
Law of Cosines ASSIGNMENT
Precalculus
1.) Solve for x
5
x
35°
6
2.) Solve for the angle (in degrees)
x°
5
6
7
For 3 and 4 solve for each triangle (meaning find all side lengths and angles).
Give lengths to three significant digits and angle measures to the nearest tenth
of a degree.
3.) a = 8, b = 5, C = 600
c = ______
A = ______
B = _______
4.) a = 8, b = 7, c = 13
A = ______
B = _______
C = ______
5.) Use the method of Example 3 to find AD in the diagram below (the figure is
not necessarily drawn to scale).
A
AB = 8, BD = 7, DC = 5, AC = 10
B
C
D
AD = ____________
6.) Find the area of the quadrilateral to the nearest square unit.
12
10
132°
108°
10
Area = __________