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 Δ XYZ.

∠𝒁=? ∠𝐙=𝟓𝟕°

33 °

𝑌 𝑍

𝑋

𝑥

17 𝑦

𝑺𝒊𝒅𝒆 𝒙=? 𝑺𝒊𝒅𝒆 𝒙=𝟏𝟏 .𝟎𝟒tan𝟑𝟑=𝒙𝟏𝟕

𝑺𝒊𝒅𝒆 𝒚=? 𝑺𝒊𝒅𝒆 𝒚=𝟐𝟎 .𝟐𝟕cos𝟓𝟕=𝟏𝟏 .𝟎𝟒

𝒚

𝟓𝟕°𝟏𝟏 .𝟎𝟒

𝟐𝟎 .𝟐𝟕

Warm-Up

Solve Δ ABC.

∠𝑨=? ∠𝐀=𝟒𝟗°

4 1°𝐶 𝐵

𝐴

𝑎

22𝑏

𝑺𝒊𝒅𝒆𝒂=? 𝑺𝒊𝒅𝒆𝒂=𝟏𝟔 .𝟔𝟎𝟒sin𝟒𝟗=𝒂𝟐𝟐

𝑺𝒊𝒅𝒆𝒃=? 𝑺𝒊𝒅𝒆𝒃=𝟏𝟒 .𝟒𝟑𝟑cos𝟒𝟗=𝒃𝟐𝟐

𝟒𝟗°

𝟏𝟔 .𝟔𝟎𝟒

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 94𝑎

1. Draw a picture

4. Write the Law of Sine proportion

sin 1925

=sin 67𝑐

𝐵

𝐶𝐴 2 5

𝟕𝟔 .𝟔𝟎𝟐𝟏𝟗°94 ° 67 °

𝟕𝟎 .𝟔𝟖𝟒

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

=sin𝐵

42

2. Draw a picture

5. Write the Law of Sine proportion

sin 12763

=sin 20.831

𝑐

𝐵

𝐶𝐴

63𝟑𝟐 .𝟏𝟔𝟗°

𝟐𝟎 .𝟖𝟑𝟏°

127 °42

4. Find the third angle,

𝟐𝟖 .𝟎𝟓𝟐

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

𝐵

𝐴

6.552 °

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 𝐴6.5

, a h

Solve the SSA case with two solutionsEXAMPLE 4Solve ABC with and .

SOLUTION

3. Write the Law of Sine proportion

sin 6232

=sin𝐵

34

2. Draw a picture

5. Write the Law of Sine proportion

sin 6 232

=sin 48.26

𝑐

𝐵

𝐶𝐴

𝟐𝟕 .𝟎𝟒𝟑69

𝟒𝟖 .𝟐𝟔°

62 °

3 2

4. Find the third angle,

1. Evaluate SSA case, a b

𝐵 𝐶𝐴

62 °3 2

3 4 3 4

, a h

Two possible triangles

𝟏𝟏𝟎 .𝟐𝟔°

𝟒𝟖 .𝟐𝟔°

𝟕 .𝟕𝟒°4. Find the third angle,

5. Write the Law of Sine proportion

sin 6 232

=sin 7.74

𝑐

𝟒 .𝟖𝟖𝟏

Vocabulary

Write and Solve a real – 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 62° angle between these two sides, what will be the area of the garden?

The area of Esther’s and Frederick’s vegetable garden is 72.843.

1. Draw a picture

2. Write an equation,

SOLUTION

Lesson 9.5 Homework:Practice BPractice C “Honors”