Transcript Warm-Up - TeacherWeb

Lesson 9.5
Apply the Law of Sines
Standard Accessed: Students will prove, apply, and
model trigonometric functions and ratios.
Warm-Up
Lesson Presentation
Lesson Quiz
Warm-Up
Solve Δ XYZ.
𝑋
17
𝟐𝟎. 𝟐𝟕
𝑦
𝑌
𝟓𝟕°
𝑥 𝟏𝟏. 𝟎𝟒
33°
∠𝒁 = ?
𝑺𝒊𝒅𝒆 𝒙 = ?
𝑺𝒊𝒅𝒆 𝒚 = ?
𝑍
∠𝐙 = 𝟓𝟕°
𝒙
tan 𝟑𝟑 =
𝑺𝒊𝒅𝒆 𝒙 = 𝟏𝟏. 𝟎𝟒
𝟏𝟕
𝟏𝟏. 𝟎𝟒
𝑺𝒊𝒅𝒆 𝒚 = 𝟐𝟎. 𝟐𝟕
cos 𝟓𝟕 =
𝒚
Warm-Up
Solve Δ ABC.
𝐴
𝑏 𝟒𝟗°
14. 𝟒𝟑𝟑
𝐶
∠𝑨 = ?
𝑺𝒊𝒅𝒆 𝒂 = ?
𝑺𝒊𝒅𝒆 𝒃 = ?
22
41°
𝑎 𝟏𝟔. 𝟔𝟎𝟒 𝐵
∠𝐀 = 𝟒𝟗°
𝒂
sin 𝟒𝟗 =
𝟐𝟐
𝒃
cos 𝟒𝟗 =
𝟐𝟐
𝑺𝒊𝒅𝒆 𝒂 = 𝟏𝟔. 𝟔𝟎𝟒
𝑺𝒊𝒅𝒆 𝒃 = 𝟏𝟒. 𝟒𝟑𝟑
Vocabulary
Can be used to solve triangles
with no right angle, when two
angles and the length of any side
are known (AAS or ASA cases), or
when the lengths of two sides and
an angle opposite one of the two
sides are known, (SSA).
Essential Understandings
When can the Law of sines be used to
solve a triangle?
 When two angles and the length of any
side are known (AAS or ASA cases), or
when the lengths of two sides and an
angle opposite one of the two sides are
known, (SSA). The law of sines is used to
solve triangles with no right angle.
EXAMPLE
1
Solve a triangle for the AAS or ASA case
Solve ΔABC with 𝐴 = 94°, 𝐶 = 67°, and 𝑏 = 25.
𝐴
SOLUTION
1. Draw a picture 
94°
𝟕𝟎. 𝟔𝟖𝟒 𝟏𝟗°
25
𝐶
67°
𝟕𝟔. 𝟔𝟎𝟐
𝐵
2. Find the third angle, 180 − 94 + 67 = 𝟏𝟗°
sin 19 sin 94
3. Write the Law of Sine proportion
=
25
𝑎
sin 19 sin 67
4. Write the Law of Sine proportion
=
25
𝑐
∴In ABC, ∠𝑩 = 𝟏𝟗°, 𝒂 ≈ 𝟕𝟔. 𝟔𝟎𝟐, and c≈ 𝟕𝟎. 𝟔𝟖𝟒.
EXAMPLE
1
Solve a triangle for the AAS or ASA case
Key Question
When you are given the measures of two angles and one side of
a triangle, why does it not matter whether the given side is the
one included between the two angles?
It does not matter since you are given the measures of any
two angles, you can always find the third angle measure.
Essential Understandings
What are the possible triangles for the SSA case?
 ∠𝐴 is obtuse & a > b, one triangle
 ∠𝐴 is acute & a = height, one triangle
 ∠𝐴 is acute & a ≥ 𝑏, one triangle
 ∠𝐴 is acute & b > a > h, Two triangles
Note: ∠𝐴 is always across from side a
Note: To find height = 𝑏 sin 𝐴
EXAMPLE
2
Solve the SSA case with one solution
Solve ΔABC with 𝐴 = 127°, 𝑎 = 63, and 𝑏 = 42.
𝐴
SOLUTION
2. Draw a picture 
𝟐𝟖. 𝟎𝟓𝟐
𝐵
42
127°
𝐶
𝟐𝟎. 𝟖𝟑𝟏°
63
𝟑𝟐. 𝟏𝟔𝟗°
1. Evaluate SSA case, a > b & ∠𝐴 is obtuse, ∴ One triangle
3. Write the Law of Sine proportion
sin 127 sin 𝐵
=
63
42
4. Find the third angle, 180 − 127 + 32.169 = 𝟐𝟎. 𝟖𝟑𝟏°
sin 127 sin 20.831
5. Write the Law of Sine proportion
=
63
𝑐
∴In ABC, ∠𝑩 ≈ 𝟑𝟐. 𝟏𝟔𝟗°, 𝒂∠𝑪 ≈ 𝟐𝟎. 𝟖𝟑𝟏, and c≈ 𝟐𝟖. 𝟎𝟓𝟐.
EXAMPLE
3
Examine the SSA case with no solution
Solve ΔABC with B = 52°, 𝑎 = 6.5, and 𝑏 = 4.7
𝐴
4.7
SOLUTION
52°
2. Draw a picture 
𝐵
1. Evaluate SSA case, a < b
ℎ = 6.5 sin 52 = 5.122, a < h
6.5
∴No solution (The triangle is not possible.)
What will happen if you try to solve the triangle by applying
the law of sines and using your calculator?
sin 52 sin 𝐴
Write the Law of Sine proportion
=
4.7
6.5
Trying to find sin−𝟏 𝟏. 𝟎𝟗 will give an ERROR message, since
the value of the sine function is never greater than 1.
EXAMPLE
4
Solve the SSA case with two solutions
Solve ΔABC with 𝐴 = 62°, 𝑎 = 32, and 𝑏 = 34.
SOLUTION
69. 𝟕𝟒°
𝟐𝟕. 𝟎𝟒𝟑
2. Draw a picture 
𝐴
𝐵
32
𝟒𝟖. 𝟐𝟔°
62°
34
𝟏𝟏𝟎. 𝟐𝟔°
32
𝐵
𝐶 𝟒. 𝟖𝟖𝟏 62°
34
𝐴
1. Evaluate SSA case, a < b ℎ = 34 sin 62 = 30.02, a > h
∴ Two possible triangles
3. Write the Law of Sine proportion
sin 62 sin 𝐵
=
32
34
4. Find the third angle, 180 − 62 + 69.74 = 𝟒𝟖. 𝟐𝟔°
4. Find the third angle, 180 − 62 + 110.26 = 𝟕. 𝟕𝟒°
5. Write the Law of Sine proportion
5. Write the Law of Sine proportion
sin 62 sin 48.26
=
32
𝑐
sin 62 sin 7.74
=
32
𝑐
𝟕. 𝟕𝟒°
𝐶
Vocabulary
EXAMPLE
4
Write and Solve a real – Use trig to find area of triangle
KIS Vegetable Garden Esther and Frederick are setting up
a triangular vegetable garden at KIS. They lay off lengths of
15ft and 11ft for two sides of the garden with a 62° angle
between these two sides, what will be the area of the
garden?
SOLUTION
1. Draw a picture 
1
2
2. Write an equation, 𝑎𝑏 sin 62
∴The area of Esther’s and Frederick’s
vegetable garden is ≈72.843𝒇𝒕𝟐 .
Lesson 9.5 Homework:
Practice B
Practice C “Honors”