law of sines · 2017. 10. 27. · day 1 law of sines notes.notebook 6 october 27, 2017 using the...
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Day 1 law of sines notes.notebook
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October 27, 2017
Solving Triangles Review
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Law of Sines
or
These equations are used to solve oblique triangles. You must be given the following to
use this law.
1) Two angles and any side (AAS or ASA)2) Two sides and an angle opposite one of them (SSA)* This second case is an ambiguous case
A
B
C
a
b
c
* To label a triangle, the vertices are labeled with capital letters and the sides are labeled with lower case letters. Side 'a' is opposite angle 'A'. This follows for the other sides and angles also.
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A
B
C
a
b
c
b = 27.4 ft.C = 102.3oB = 28.7o
27.4 ft.
102.3o
28.7o
Solve the triangle: (this means to find all the missing side lengths and angles)
Set up a proportion using the given information and one missing piece:
Use cross products to solve the proportion:
You can find the missing angle by subtracting the given angles from 180o.
ft
m A =180 28.7 102.3 = 490
You need to use this angle to find the last side length a.
a = 43.06 ft
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Solve the triangle with the given information:
A = 43o B = 98o c = 22 ft.
AB
C
AB
C
98o 43o
22 ft.
23.84 ft
39o34.6 ft
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Using the sine function to find the AREA of an oblique triangle.
Area = 1/2 bc sin A = 1/2 ab sin C = 1/2 ac sin B
Use these to find the area of each triangle.
A B
C
57o
22 mm
32 mm
B
AC79o
15 ft. 25 ft.
(you need to find more information before you can use the area formula)
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The AMBIGUOUS CASE (SSA)
For a given triangle: a = 20 in. b = 25 in. A = 40o
A B
C
A B
C20 in
20 in25 in 25 in40o 40o
2 possible pictures
If B = 53.46o If B = 126.54o
c ≈ 7.24
* Key Step
* Rule for possible ambiguous situations:
At the key step, find your angle using the inverse sine (sin1) function on your calculator. After you find that angle, subtract it from 180o to find the other possible angle with the same sine value. Add this angle with the other angle you know. If these two angles have a sum less than 180o, then you will have 2 possible solutions for your triangle.
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a = 10, b = 16, and m<A = 30º.
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a = 20, c = 16, and m<A = 30º
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a = 7, c = 16, and m<A = 30º.
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LAW OF SINES
1. You are standing 40 meters from the base of a tree that is leaning 8° from the vertical away from you. The angle of elevation from your feet to the top of the tree is 20°50'. *FIND THE SLANT HEIGHT OF THE TREE.
A B
C
20°50' 90°8°
40 m
s(slant height of tree)
s = 16.24 meters
2. A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E. Find the distance from the boat the shoreline if the lighthouse is at the shoreline.
A B
C D(shoreline)
(.25 × 10) = 2.5 miles
70°63°
d = 3.18 miles
d
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