Solving Triangles with Law of Sines
Warm – Up
Solve the following triangles for the missing side or angle:
1)
2)
3)
32°
48°
9
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Law of Sines
Section 6.1
Objectives
Students will be able to…
Solve oblique triangles using the Law of
Sines (including the ambiguous case)
Solve application problems for all triangles
using trigonometry
Homework
Law of Sines Worksheet - EVENS
Oblique Triangles
Triangles without a right angle
Four Cases:
1.
Two angles and any side (AAS or ASA)
2.
Two sides and an angle opposite one of
them (SSA) (ambiguous case)
3.
Three sides (SSS)
4.
Two sides and their included angle (SAS)
First
2
cases can be solved using the
Law of Sines
Law Of Sines
Example - AAS
Given C = 102.3°, B = 28.7° and b = 27.4 feet, find the remaining angles and sides
Example - ASA
A pole tilts toward the sun at an 8° angle from the vertical and it casts a 22 foot
shadow. The angle of elevation from the tip of the shadow to the top of the pole
is 43°. How tall is the pole?
Ambiguous Case - SSA
In a triangle where you are given
a
,
b
and
h
(h =
b
sin
A
)
If angle
A
is acute
And
a
<
h
– no triangle is possible
And
a
=
h
– one triangle is possible (right triangle)
And
a
≥
b
– one triangle is possible
And
h
<
a
<
b
– two triangles are possible
If angle
A
is obtuse
And
a
≤
b
– no triangle is possible
And
a
>
b
– one triangle is possible
Example
Given
a
= 15,
b
= 25, and
A
= 85°, solve the triangle.
Example
Given
a
= 22 in,
b
= 12 in, and
A
= 42°, find the remaining
side and angles.
Example
Solve the triangle, given
a
= 12 m,
b
= 31 m, and
A
= 20.5°.
Example
A bridge is to be built across a small lake from a gazebo to the dock is S 41° W.
From a tree 100m from the gazebo, the bearings to the gazebo and the dock are
S 74° E and S 28° E respectively. Find the distance from the gazebo to the dock.
Practice
Law of Sines Worksheet
Closure
Given
A
= 102.4°,
C
= 16.7°, and
a
= 21.6, solve for
the remaining angle and sides
Learn how to solve oblique triangles using the Law of Sines, including the ambiguous case. Practice finding missing sides and angles in triangles without right angles. Explore examples and applications to enhance your trigonometry skills.
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Presentation Transcript
Warm Up Solve the following triangles for the missing side or angle: 1) 2) 3) 9 10 32 48 14 27 8 x x
Law of Sines Section 6.1
Objectives Students will be able to Solve oblique triangles using the Law of Sines (including the ambiguous case) Solve application problems for all triangles using trigonometry
Homework Law of Sines Worksheet - EVENS
Oblique Triangles Triangles without a right angle Four Cases: C 1. Two angles and any side (AAS or ASA) a b 2. Two sides and an angle opposite one of them (SSA) (ambiguous case) A B c 3. Three sides (SSS) 4. Two sides and their included angle (SAS) First 2 cases can be solved using the Law of Sines
Law Of Sines If ABC is a triangle with sides a, b, and c, then ? ????= ? ? ????= ???? C C a b a b 0 0 A c B B c A *Can be written as ???? =???? =???? ? ? ?
Example - AAS Given C = 102.3 , B = 28.7 and b = 27.4 feet, find the remaining angles and sides C a b A B c
Example - ASA A pole tilts toward the sun at an 8 angle from the vertical and it casts a 22 foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43 . How tall is the pole? 0
Ambiguous Case - SSA In a triangle where you are given a, b and h (h = b sinA) If angle A is acute And a < h no triangle is possible And a = h one triangle is possible (right triangle) And a b one triangle is possible And h < a < b two triangles are possible If angle A is obtuse And a b no triangle is possible And a > b one triangle is possible
Example Given a = 15, b = 25, and A = 85 , solve the triangle.
Example Given a = 22 in, b = 12 in, and A = 42 , find the remaining side and angles.
Example Solve the triangle, given a = 12 m, b = 31 m, and A = 20.5 .
Example A bridge is to be built across a small lake from a gazebo to the dock is S 41 W. From a tree 100m from the gazebo, the bearings to the gazebo and the dock are S 74 E and S 28 E respectively. Find the distance from the gazebo to the dock. T G D
Practice Law of Sines Worksheet
Closure Given A = 102.4 , C = 16.7 , and a = 21.6, solve for the remaining angle and sides
