holt mcdougal algebra 2 10-5 the law of sines 10-5 the law of sines holt algebra 2 warm up warm up...
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Holt McDougal Algebra 2
10-5 The Law of Sines10-5 The Law of Sines
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
10-5 The Law of Sines
Warm Up
Find the area of each triangle with the given base and height.
1. b =10, h = 7 2. b = 8, h = 4.6
35 units2 18.4 units2
Solve each proportion.
3. 4.
In ∆ABC, mA = 122° and mB =17°. What is the mC ?
5.
28.5 10
41°
Holt McDougal Algebra 2
10-5 The Law of Sines
Determine the area of a triangle given side-angle-side information.
Use the Law of Sines to find the side lengths and angle measures of a triangle.
Objectives
Holt McDougal Algebra 2
10-5 The Law of Sines
The area of the triangle representing the sail is Although you do not know the value of h, you can calculate it by using the fact that sin A = , or h = c sin A.
A sailmaker is designing a sail that will have the dimensions shown in the diagram. Based on these dimensions, the sailmaker can determine the amount of fabric needed.
Holt McDougal Algebra 2
10-5 The Law of Sines
Area =
Area =
This formula allows you to determine the area of a triangle if you know the lengths of two of its sides and the measure of the angle between them.
Write the area formula.
Substitute c sin A for h.
Holt McDougal Algebra 2
10-5 The Law of Sines
An angle and the side opposite that angle are labeled with the same letter. Capital letters are used for angles, and lowercase letters are used for sides.
Helpful Hint
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 1: Determining the Area of a Triangle
Find the area of the triangle. Round to the nearest tenth.
Area = ab sin C
≈ 4.820907073
Write the area formula.
Substitute 3 for a, 5 for b, and 40° for C.
Use a calculator to evaluate the expression.
The area of the triangle is about 4.8 m2.
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 1
Find the area of the triangle. Round to the nearest tenth.
Area = ac sin B
≈ 47.88307441
Write the area formula.
Substitute 8 for a, 12 for c, and 86° for B.
Use a calculator to evaluate the expression.
The area of the triangle is about 47.9 m2.
Holt McDougal Algebra 2
10-5 The Law of Sines
The area of ∆ABC is equal to bc sin A or ac sin B or ab sin C. By setting these expressions equal to each other, you can derive the Law of Sines.
bc sin A = ac sin B = ab sin C
bc sin A = ac sin B = ab sin C
bc sin A ac sin B ab sin C abc abc abc
= =
sin A = sin B = sin C a b c
Multiply each expression by 2.
Divide each expression by abc.
Divide out common factors.
Holt McDougal Algebra 2
10-5 The Law of Sines
The Law of Sines allows you to solve a triangle as long as you know either of the following:
1. Two angle measures and any side length–angle-angle-side (AAS) or angle-side-angle (ASA) information
2. Two side lengths and the measure of an angle that is not between them–side-side-angle (SSA) information
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 2A: Using the Law of Sines for AAS and ASA
Solve the triangle. Round to the nearest tenth.
Step 1. Find the third angle measure.
mD + mE + mF = 180°
33° + mE + 28° = 180°
mE = 119°
Triangle Sum Theorem.
Substitute 33° for mD and 28° for mF.
Solve for mE.
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 2A Continued
Step 2 Find the unknown side lengths.
sin D sin Fd f
=sin E sin F
e f=
sin 33° sin 28°d 15=
sin 119° sin 28°e 15=
d sin 28° = 15 sin 33° e sin 28° = 15 sin 119°
d = 15 sin 33°sin 28°
d ≈ 17.4
e = 15 sin 119°sin 28°
e ≈ 27.9Solve for the
unknown side.
Law of Sines.
Substitute.
Crossmultiply.
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 2B: Using the Law of Sines for AAS and ASA
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
mP = 180° – 36° – 39° = 105° Triangle Sum Theorem
Q
r
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 2B: Using the Law of Sines for AAS and ASA
Solve the triangle. Round to the nearest tenth.
Step 2 Find the unknown side lengths.
sin P sin Qp q= sin P sin R
p r=Law of Sines.
sin 105° sin 36°10 q= sin 105° sin 39°
10 r=Substitute.
q = 10 sin 36°sin 105°
≈ 6.1 r = 10 sin 39°sin 105°
≈ 6.5
Q
r
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 2a
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
mK = 31° Solve for mK.
mH + mJ + mK = 180°
42° + 107° + mK = 180°Substitute 42° for mH
and 107° for mJ.
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 2a Continued
Step 2 Find the unknown side lengths.
sin H sin Jh j
=sin K sin H
k h=
sin 42° sin 107°h 12=
sin 31° sin 42°k 8.4=
h sin 107° = 12 sin 42° 8.4 sin 31° = k sin 42°
h = 12 sin 42°sin 107°
h ≈ 8.4
k = 8.4 sin 31°sin 42°
k ≈ 6.5Solve for the
unknown side.
Law of Sines.
Substitute.
Crossmultiply.
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 2b
Solve the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
mN = 180° – 56° – 106° = 18° Triangle Sum Theorem
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 2b
Solve the triangle. Round to the nearest tenth.
Step 2 Find the unknown side lengths.
sin N sin Mn m= sin M sin P
m p=Law of Sines.
Substitute.sin 18° sin 106°1.5 m=
m = 1.5 sin 106°sin 18°
≈ 4.7 p = 4.7 sin 56°sin 106°
≈ 4.0
sin 106° sin 56°4.7 p=
Holt McDougal Algebra 2
10-5 The Law of Sines
When you use the Law of Sines to solve a triangle for which you know side-side-angle (SSA) information, zero, one, or two triangles may be possible. For this reason, SSA is called the ambiguous case.
Holt McDougal Algebra 2
10-5 The Law of Sines
When one angle in a triangle is obtuse, the measures of the other two angles must be acute.
Remember!
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 3: Art Application
Determine the number of triangular banners that can be formed using the measurements a = 50, b = 20, and mA = 28°. Then solve the triangles. Round to the nearest tenth.
Step 1 Determine the number of possible triangles. In this case, A is acute.
Because b < a; only one triangle is possible.
A B
C
b a
c
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 3 Continued
Step 2 Determine mB.
Law of Sines
Substitute.
Solve for sin B.
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 3 Continued
Let B represent the acute angle with a sine of 0.188. Use the inverse sine function on your calculator to determine mB.
Step 3 Find the other unknown measures of the triangle.
Solve for mC.
28° + 10.8° + mC = 180°
mC = 141.2°
m B = Sin-1
Holt McDougal Algebra 2
10-5 The Law of Sines
Example 3 Continued
Solve for c.
c ≈ 66.8
Law of Sines
Substitute.
Solve for c.
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 3
Determine the number of triangles Maggie can form using the measurements a = 10 cm, b = 6 cm, and mA =105°. Then solve the triangles. Round to the nearest tenth.
Step 1 Determine the number of possible triangles. In this case, A is obtuse.
Because b < a; only one triangle is possible.
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 3 Continued
Step 2 Determine mB.
Law of Sines
Substitute.
Solve for sin B.
sin B ≈ 0.58
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 3 Continued
m B = Sin-1
Let B represent the acute angle with a sine of 0.58. Use the inverse sine function on your calculator to determine m B.
Step 3 Find the other unknown measures of the triangle.
Solve for mC.
105° + 35.4° + mC = 180°
mC = 39.6°
Holt McDougal Algebra 2
10-5 The Law of Sines
Check It Out! Example 3 Continued
Solve for c.
c ≈ 6.6 cm
Law of Sines
Substitute.
Solve for c.
Holt McDougal Algebra 2
10-5 The Law of Sines
Lesson Quiz: Part I
1. Find the area of the triangle. Round to the nearest tenth.
17.8 ft2
2. Solve the triangle. Round to the nearest tenth.
a 32.2; b 22.0; mC = 133.8°
Holt McDougal Algebra 2
10-5 The Law of Sines
Lesson Quiz: Part II
3. Determine the number of triangular quilt pieces that can be formed by using the measurements a = 14 cm, b = 20 cm, and mA = 39°. Solve each triangle. Round to the nearest tenth. 2;
c1 21.7 cm;mB1 ≈ 64.0°;mC1 ≈ 77.0°;
c2 ≈ 9.4 cm;mB2 ≈ 116.0°;mC2 ≈ 25.0°