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13-6. The Law of Cosines. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Find each measure to the nearest tenth. 1 . m y 2. x 3. y 4. What is the area of ∆ XYZ ? Round to the nearest square unit. ≈ 8.8. 104°. ≈ 18.3. 60 square units. Objectives. - PowerPoint PPT Presentation
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Holt Algebra 2
13-6 The Law of Cosines13-6 The Law of Cosines
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
13-6 The Law of Cosines
Warm Up
Find each measure to the nearest tenth.
1. my 2. x
3. y
4. What is the area of ∆XYZ ? Round to the
nearest square unit. 60 square units
104° ≈ 8.8
≈ 18.3
Holt Algebra 2
13-6 The Law of Cosines
Use the Law of Cosines to find the side lengths and angle measures of a triangle.
Use Heron’s Formula to find the area of a triangle.
Objectives
Holt Algebra 2
13-6 The Law of Cosines
In the previous lesson, you learned to solve triangles by using the Law of Sines. However, the Law of Sines cannot be used to solve triangles for which side-angle-side (SAS) or side-side-side (SSS) information is given. Instead, you must use the Law of Cosines.
Holt Algebra 2
13-6 The Law of Cosines
To derive the Law of Cosines, draw ∆ABC with altitude BD. If x represents the length of AD, the length of DC is b – x.
Holt Algebra 2
13-6 The Law of Cosines
Write an equation that relates the side lengths of ∆DBC.
a2 = (b – x)2 + h2
a2 = b2 – 2bx + x2 + h2
a2 = b2 – 2bx + c2
a2 = b2 – 2b(c cos A) + c2
a2 = b2 + c2 – 2bccos A
Pythagorean Theorem
Expand (b – x)2.
In ∆ABD, c2 = x2 + h2. Substitute c2 for x2 + h2.
The previous equation is one of the formulas for the Law of Cosines.
In ∆ABD, cos A = or x = cos A. Substitute c cos A for x.
Holt Algebra 2
13-6 The Law of Cosines
Holt Algebra 2
13-6 The Law of Cosines
Example 1A: Using the Law of Cosines
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
a = 8, b = 5, mC = 32.2°
Step 1 Find the length of the third side.
c2 = a2 + b2 – 2ab cos C
c2 = 82 + 52 – 2(8)(5) cos 32.2°
c2 ≈ 21.3
c ≈ 4.6
Law of Cosines
Substitute.
Use a calculator to simplify.
Solve for the positive value of c.
Holt Algebra 2
13-6 The Law of Cosines
Step 2 Find the measure of the smaller angle, B.
Law of Sines
Substitute.
Solve for sin B.
Solve for m B.
Example 1A Continued
Holt Algebra 2
13-6 The Law of Cosines
Example 1A Continued
Step 3 Find the third angle measure.
mA + 35.4° + 32.2° 180°
mA 112.4°
Triangle Sum Theorem
Solve for m A.
Holt Algebra 2
13-6 The Law of Cosines
Example 1B: Using the Law of Cosines
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
a = 8, b = 9, c = 7
Step 1 Find the measure of the largest angle, B.
b2 = a2 + c2 – 2ac cos B
92 = 82 + 72 – 2(8)(7) cos B
cos B = 0.2857
m B = Cos-1 (0.2857) ≈ 73.4°
Law of cosines
Substitute.
Solve for cos B.
Solve for m B.
Holt Algebra 2
13-6 The Law of Cosines
Example 1B Continued
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
Step 2 Find another angle measure
c2 = a2 + b2 – 2ab cos C Law of cosines
72 = 82 + 92 – 2(8)(9) cos C Substitute.
cos C = 0.6667 Solve for cos C.
m C = Cos-1 (0.6667) ≈ 48.2° Solve for m C.
Holt Algebra 2
13-6 The Law of Cosines
Example 1B Continued
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
m A + 73.4° + 48.2° 180°
m A 58.4°
Triangle Sum Theorem
Solve for m A.
Step 3 Find the third angle measure.
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 1a
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
Step 1 Find the length of the third side.
a2 = b2 + c2 – 2bc cos A
a2 = 232 + 182 – 2(23)(18) cos 173°
a2 ≈ 1672.8
a ≈ 40.9
Law of Cosines
Substitute.
Use a calculator to simplify.
Solve for the positive value of c.
b = 23, c = 18, m A = 173°
Holt Algebra 2
13-6 The Law of Cosines
Law of Sines
Substitute.
Solve for sin C.
Solve for m C.
Step 2 Find the measure of the smaller angle, C.
m C = Sin-1
Check It Out! Example 1a Continued
Holt Algebra 2
13-6 The Law of Cosines
Step 3 Find the third angle measure.
m B + 3.1° + 173° 180°
m B 3.9°
Triangle Sum Theorem
Solve for m B.
Check It Out! Example 1a Continued
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 1b
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
a = 35, b = 42, c = 50.3
Step 1 Find the measure of the largest angle, C.
c2 = a2 + b2 – 2ab cos C
50.32 = 352 + 422 – 2(35)(50.3) cos C
cos C = 0.1560
m C = Cos-1 (0.1560) ≈ 81.0°
Law of cosines
Substitute.
Solve for cos C.
Solve for m C.
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 1b Continued
Step 2 Find another angle measure
a2 = c2 + b2 – 2cb cos A Law of cosines
352 = 50.32 + 422 – 2(50.3)(42) cos A Substitute.
cos A = 0.7264 Solve for cos A.
m A = Cos-1 (0.7264) ≈ 43.4° Solve for m A.
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
a = 35, b = 42, c = 50.3
Holt Algebra 2
13-6 The Law of Cosines
Step 3 Find the third angle measure.
m B + 81° + 43.4° 180°
m B 55.6° Solve for m B.
Check It Out! Example 1b Continued
Holt Algebra 2
13-6 The Law of Cosines
The largest angle of a triangle is the angle opposite the longest side.
Remember!
Holt Algebra 2
13-6 The Law of Cosines
Example 2: Problem-Solving Application
If a hiker travels at an average speed of 2.5 mi/h, how long will it take him to travel from the cave to the waterfall? Round to the nearest tenth of an hour.
11 Understand the Problem
The answer will be the number of hours that the hiker takes to travel to the waterfall.
Holt Algebra 2
13-6 The Law of Cosines
List the important information:
• The cave is 3 mi from the cabin.
• The waterfall is 4 mi from the cabin. The path from the cabin to the waterfall makes a 71.7° angle with the path from the cabin to the cave.
• The hiker travels at an average speed of 2.5 mi/h.
11 Understand the Problem
The answer will be the number of hours that the hiker takes to travel to the waterfall.
Holt Algebra 2
13-6 The Law of Cosines
22 Make a Plan
Use the Law of Cosines to find the distance d between the water-fall and the cave. Then determine how long it will take the hiker to travel this distance.
Holt Algebra 2
13-6 The Law of Cosines
d2 = c2 + w2 – 2cw cos D
d2 = 42 + 32 – 2(4)(3)cos 71.7°
d2 ≈ 17.5
d ≈ 4.2
Law of Cosines
Substitute 4 for c, 3 for w, and 71.7 for D.
Use a calculator to simplify.
Solve for the positive value of d.
Solve33
Step 1 Find the distance d between the waterfall and the cave.
Holt Algebra 2
13-6 The Law of Cosines
Step 2 Determine the number of hours.
The hiker must travel about 4.2 mi to reach the waterfall. At a speed of 2.5 mi/h, it will take the hiker ≈ 1.7 h to reach the waterfall.
Look Back44
In 1.7 h, the hiker would travel 1.7 2.5 = 4.25 mi. Using the Law of Cosines to solve for the angle gives 73.1°. Since this is close to the actual value, an answer of 1.7 hours seems reasonable.
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 2
A pilot is flying from Houston to Oklahoma City. To avoid a thunderstorm, the pilot flies 28° off the direct route for a distance of 175 miles. He then makes a turn and flies straight on to Oklahoma City. To the nearest mile, how much farther than the direct route was the route taken by the pilot?
Holt Algebra 2
13-6 The Law of Cosines
11 Understand the Problem
The answer will be the additional distance the pilot had to fly to reach Oklahoma City.
List the important information:
• The direct route is 396 miles.
• The pilot flew for 175 miles off the course at an angle of 28° before turning towards Oklahoma City.
Holt Algebra 2
13-6 The Law of Cosines
22 Make a Plan
Use the Law of Cosines to find the distance from the turning point on to Oklahoma City. Then determine the difference additional distance and the direct route.
Holt Algebra 2
13-6 The Law of Cosines
b2 = c2 + a2 – 2ca cos B
b2 = 3962 + 1752 – 2(396)(175)cos 28°
b2 ≈ 65072
b ≈ 255
Law of Cosines
Substitute 396 for c, 175 for a, and 28° for B.
Use a calculator to simplify.
Solve for the positive value of b.
Solve33
Step 1 Find the distance between the turning point and Oklahoma City. Use side-angle-side.
Holt Algebra 2
13-6 The Law of Cosines
255 + 175 = 430
430 – 396 = 34
Total miles traveled.
Additional miles.
Step 2 Determine the number of additional miles the plane will fly.
Add the actual miles flown and subtract from that normal distance to find the extra miles flown
Look Back44
By using the Law of Cosines the length of the extra leg of the trip could be determined.
Holt Algebra 2
13-6 The Law of Cosines
The Law of Cosines can be used to derive a formula for the area of a triangle based on its side lengths. This formula is called Heron’s Formula.
Holt Algebra 2
13-6 The Law of Cosines
Example 3: Landscaping Application
A garden has a triangular flower bed with sides measuring 2 yd, 6 yd, and 7 yd. What is the area of the flower bed to the nearest tenth of a square yard?
Step 1 Find the value of s.
Use the formula for half of the perimeter.
Substitute 2 for a, 6 for b, and 7 for c.
Holt Algebra 2
13-6 The Law of Cosines
Example 3 Continued
Step 2 Find the area of the triangle.
A =
A =
A = 5.6
Heron’s formula
Substitute 7.5 for s.
Use a calculator to simplify.
The area of the flower bed is 5.6 yd2.
Holt Algebra 2
13-6 The Law of Cosines
Example 3 Continued
Check Find the measure of the largest angle, C.
c2 = a2 + b2 – 2ab cos C
72 = 22 + 62 – 2(2)(6) cos C
cos C ≈ –0.375
m C ≈ 112.0°
Find the area of the triangle by using the formula area = ab sin c.
area
Law of Cosines
Substitute.
Solve for cos C.
Solve for m c.
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 3
The surface of a hotel swimming pool is shaped like a triangle with sides measuring 50 m, 28 m, and 30 m. What is the area of the pool’s surface to the nearest square meter?
Step 1 Find the value of s.
Use the formula for half of the perimeter.
Substitute 50 for a, 28 for b, and 30 for c.
Holt Algebra 2
13-6 The Law of Cosines
Step 2 Find the area of the triangle.
A =
A = 367 m2
Heron’s formula
Substitute 54 for s.
Use a calculator to simplify.
The area of the flower bed is 367 m2.
Check It Out! Example 3 Continued
Holt Algebra 2
13-6 The Law of Cosines
Check It Out! Example 3 Continued
Check Find the measure of the largest angle, A.
502 = c2 + b2 – 2cb cos A
502 = 302 + 282 – 2(30)(28) cos A
cos A ≈ –0.4857
m A ≈ 119.0°
Find the area of the triangle by using the formula area = ab sin A.
Law of Cosines
Substitute.
Solve for cos A.
Solve for m A.
Holt Algebra 2
13-6 The Law of Cosines
Lesson Quiz: Part I
Use the given measurements to solve ∆ABC. Round to the nearest tenth.
1. a = 18, b = 40, m C = 82.5°
c ≈ 41.7; m A ≈ 25.4°; m B ≈ 72.1°
2. a = 18.0; b = 10; c = 9m A ≈ 142.6°; m B ≈ 19.7°; m C ≈ 17.7°
Holt Algebra 2
13-6 The Law of Cosines
Lesson Quiz: Part II
3. Two model planes take off from the same spot. The first plane travels 300 ft due west before landing and the second plane travels 170 ft southeast before landing. To the nearest foot, how far apart are the planes when they land?437 ft
4. An artist needs to know the area of a triangular piece of stained glass with sides measuring 9 cm, 7 cm, and 5 cm. What is the area to the nearest square centimeter?17 cm2
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