Transcript 6.2: Law of Cosines

Re:view
1. Use the Law of Sines to solve:
1) A=36º, a=10 meters, b=4 meters
2) A=24.3º, C=54.6º, c=2.68 cm
2. Solve ABC
A
5
32º
C
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6.2: Law of Cosines
Objectives:
• Use Law of Cosines to solve oblique
triangles
• Use Law of Cosines to model and
solve real-life problems
• Use Heron’s Area Formula to find
areas of Triangles
Must haves for solving
Oblique Triangles
• 2 angles and any side (AAS or ASA)
• 2 sides and an angle opposite one of them (SSA)
• 3 sides (SSS)
• 2 sides and their included angle (SAS)
• The first two cases can be solved using the Law of Sines.
• The last two cases can be solved using the
Law of Cosines
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Law of Cosines
Alternative Form
Standard Form
b c a
cos A 
2bc
b 2  a 2  c 2  2accos B
2
2
2
a c b
2
2
2
cos B 
c  a  b  2ab cos C
2ac
Given an angle (A, B, C), the side
length (a, b, c) is equal to the
a 2  b2  c2
cos C 
sum of the squares of the
2ab
remaining two side lengths minus
a  b  c  2bccos A
2
2
2
twice the product of the two side
lengths and the cosine of the
angle given. #WOW 6.2 Law of Cosines
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2
2
2
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Rules for Law of Cosines
cos   0 for 0    90 ACUTE
cos   0 for 90    180 OBTUSE
• a triangle can only have one obtuse angle.
• if the largest angle is acute, the remaining 2
angles must also be acute
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3 Sides of a Triangle (SSS)
1. Use the Law of Cosines to find the angle
opposite the longest side.
2. Use the Law of Sines to find one of the
other angles.
3. Subtract the angles found in step 1 & 2
from 180º
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Example: SSS
Given: ABC
Find: All 3 angles of the triangle
B
a = 8 ft
c = 14 ft
A
C
b = 19 ft
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2 Sides and the Included Angle (SAS)
1. Use the Law of Cosines to find the
unknown side
2. Use the Law of Sines to find one of the
missing angles
3. Find the last angle by subtracting the 2
known angles from 180°
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Example SAS
Given: ABC with A=115º
Find: The remaining 2 angles of the triangle
C
15 cm
A 10 cm
B
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Re:view
1. Write the three standard form
equations for the Law of Cosines.
2. Write the formulas for Law of Sines.
3. What is the area of a triangle?
4. How else can we re-write the area of a
triangle using trig?
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Heron’s Area Formula
Given any triangle with sides of length a, b, and c,
the area of the triangle is
Area  s(s  a)(s  b)(s  c)
(a  b  c)
where s 
2
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Example: Heron’s Formula
Find the area of the triangular region having
sides of lengths a=43 meters, b=53 meters, and
c=72 meters.
1. Solve for s.
2. Apply the side lengths and the length of
s to Heron’s formula to solve for the area
of the triangle.
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Example: SSS #2
Given: On a map, Orlando is 178 millimeters due
south of Niagra Falls. Denver is 273 millimeters
from Orlando, and Denver is 235 millimeters from
Niagra Falls.
Find: the bearing of Denver from Orlando and the
bearing from Denver to Niagra Falls.
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Example: SAS #2
Given: A plane flies 810 miles from A to B with
a bearing of N75°E. Then it flies 648 miles
from B to C with a bearing of N32°E. Draw
a diagram to represent the problem and find
the straight line distance and bearing from C
and A.
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Homework
• Check Blackboard and
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