7.3 Day 2 The Law of Cosines
May 05, 2009
7.3 The Law of Cosines
Day 2
Objective: Use the law of cosines to find measures of sides and angles in triangles.
Apr 19 ­ 9:19 AM
1
7.3 Day 2 The Law of Cosines
May 05, 2009
Law of Cosines: For any triangle with sides a, b, c and opposite angles α, β, γ, respectively, the following is true. c
α
b
γ
­1
γ = cos
β = cos
­1
α = cos
­1
c2 = b2 + a2 ­ 2ab cos γ
β
b2 = a2 + c2 ­ 2ac cos β
a
(
(
(
2
2
a + b ­ c
2ab
2
a2 + c2 ­ b2
2ac
b2 + c2 ­ a2
2bc
(
(
(
a2 = b2 + c2 ­ 2bc cos α
Apr 20 ­ 3:03 PM
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7.3 Day 2 The Law of Cosines
May 05, 2009
Example #1: The distance from home plate to the fence in dead center at the Oak Lawn Little League field is 280 feet. How far is it from the fence in dead center to third base?
Hint: The distance between the bases in Little League is 60 feet.
?
280
60
241.33 feet
Apr 20 ­ 3:29 PM
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7.3 Day 2 The Law of Cosines
May 05, 2009
Example #2: A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction of 20o off the direct heading to Barbados. The captain maintains the 15­knot speed for 10 hours, after which time the path to Barbados becomes clear of storms.
Barbados
600
20o
San Juan
Apr 20 ­ 3:29 PM
4
7.3 Day 2 The Law of Cosines
May 05, 2009
Example #2: A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction of 20o off the direct heading to Barbados. The captain maintains the 15 knot speed for 10 hours, after which time the path to Barbados becomes clear of storms.
a. Through what angle should the captain turn to head directly to Barbados?
Step 1: Find the distance traveled.
15(10) = 150 nautical miles
Step 2: Find the distance remaining.
d2 = 6002 + 1502 ­ 2(600)(150)cos20o
d = 461.9 nautical miles
Barbados
600
20o
San Juan
Step 3: Find the angle towards Barbados.
sin20o
sinS
=
461.9
600
600sin20o
sinS =
461.9
sinS = 0.444278
S = sin­1(0.444278)
S = 26.4o or 180o ­ 26.4o = 153.6o
Step 4: Find the angle the captain should turn.
180o ­ 153.6o = 26.4o
Apr 20 ­ 3:29 PM
5
7.3 Day 2 The Law of Cosines
May 05, 2009
Example #2: A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction of 20o off the direct heading to Barbados. The captain maintains the 15 knot speed for 10 hours, after which time the path to Barbados becomes clear of storms.
Barbados
b. Once the turn is made how long will it be before the ship reaches Barbados if the same 15 knot speed is maintained?
Yay...we already found the distance
to be 461.9 nautical miles. Now we have
to convert that into time at 15 knot/hour.
600
20o
San Juan
461.9 ÷ 15 = 30.8 hours
Apr 20 ­ 3:29 PM
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7.3 Day 2 The Law of Cosines
May 05, 2009
HOMEWORK
page 547 (33, 36, 38, 39, 41, 48)
Apr 20 ­ 3:56 PM
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