Lesson Quiz
Lesson Presentation
Lesson 9.5Apply the Law of Sines
Warm-Up
Standard Accessed: Students will prove, apply, and model trigonometric functions and ratios.
Warm-Up
Solve Ξ XYZ.
β π=? β π=ππΒ°
33 Β°
π π
π
π₯
17 π¦
πΊππ π π=? πΊππ π π=ππ .ππtanππ=πππ
πΊππ π π=? πΊππ π π=ππ .ππcosππ=ππ .ππ
π
ππΒ°ππ .ππ
ππ .ππ
Warm-Up
Solve Ξ ABC.
β π¨=? β π=ππΒ°
4 1Β°πΆ π΅
π΄
π
22π
πΊππ ππ=? πΊππ ππ=ππ .πππsinππ=πππ
πΊππ ππ=? πΊππ ππ=ππ .πππcosππ=πππ
ππΒ°
ππ .πππ
14
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.
Solve a triangle for the AAS or ASA caseEXAMPLE 1Solve ABC with and .
SOLUTION
In ABC, and c.
2. Find the third angle,
3. Write the Law of Sine proportion
sin 19
25=
sin 94π
1. Draw a picture
4. Write the Law of Sine proportion
sin 1925
=sin 67π
π΅
πΆπ΄ 2 5
ππ .πππππΒ°94 Β° 67 Β°
ππ .πππ
Solve a triangle for the AAS or ASA caseEXAMPLE 1
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.
Key Question
Essential Understandings
What are the possible triangles for the SSA case? is obtuse & a b, one triangle
is acute & , one triangle
is acute & , one triangle
is acute & , Two triangles
Note: is always across from side a
Note: To find
Solve the SSA case with one solutionEXAMPLE 2Solve ABC with and .
SOLUTION
In ABC, and c.
3. Write the Law of Sine proportion
sin 12763
=sinπ΅
42
2. Draw a picture
5. Write the Law of Sine proportion
sin 12763
=sin 20.831
π
π΅
πΆπ΄
63ππ .πππΒ°
ππ .πππΒ°
127 Β°42
4. Find the third angle,
ππ .πππ
1. Evaluate SSA case, a b & is obtuse, One triangle
Examine the SSA case with no solutionEXAMPLE 3Solve ABC with and
SOLUTION
No solution (The triangle is not possible.)
2. Draw a picture
π΅
π΄
6.552 Β°
1. Evaluate SSA case, a b
4.7
What will happen if you try to solve the triangle by applying the law of sines and using your calculator?
Trying to find will give an ERROR message, since the value of the sine function is never greater than
1.
Write the Law of Sine proportionsin 52
4.7=
sin π΄6.5
, a h
Solve the SSA case with two solutionsEXAMPLE 4Solve ABC with and .
SOLUTION
3. Write the Law of Sine proportion
sin 6232
=sinπ΅
34
2. Draw a picture
5. Write the Law of Sine proportion
sin 6 232
=sin 48.26
π
π΅
πΆπ΄
ππ .πππ69
ππ .ππΒ°
62 Β°
3 2
4. Find the third angle,
1. Evaluate SSA case, a b
π΅ πΆπ΄
62 Β°3 2
3 4 3 4
, a h
Two possible triangles
πππ .ππΒ°
ππ .ππΒ°
π .ππΒ°4. Find the third angle,
5. Write the Law of Sine proportion
sin 6 232
=sin 7.74
π
π .πππ
Vocabulary
Write and Solve a real β Use trig to find area of triangleEXAMPLE 4
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 62Β° angle between these two sides, what will be the area of the garden?
The area of Estherβs and Frederickβs vegetable garden is 72.843.
1. Draw a picture
2. Write an equation,
SOLUTION
Lesson 9.5 Homework:Practice BPractice C βHonorsβ