example 1 solve a triangle for the sas case solve abc with a = 11, c = 14, and b = 34°. solution...
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![Page 1: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/1.jpg)
EXAMPLE 1 Solve a triangle for the SAS case
Solve ABC with a = 11, c = 14, and B = 34°.
SOLUTION
Use the law of cosines to find side length b.
b2 = a2 + c2 – 2ac cos B
b2 = 112 + 142 – 2(11)(14) cos 34°
b2 61.7
b2 61.7 7.85
Law of cosines
Substitute for a, c, and B.
Simplify.
Take positive square root.
![Page 2: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/2.jpg)
EXAMPLE 1 Solve a triangle for the SAS case
Use the law of sines to find the measure of angle A.
sin Aa
sin Bb
=
sin A11
=sin 34°7.85
sin A =11 sin 34°
7.850.7836
A sin –1 0.7836 51.6°
Law of sines
Substitute for a, b, and B.
Multiply each side by 11 andSimplify.
Use inverse sine.
The third angle C of the triangle is C 180° – 34° – 51.6° = 94.4°.
In ABC, b 7.85, A 51.68, and C 94.48.ANSWER
![Page 3: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/3.jpg)
EXAMPLE 2 Solve a triangle for the SSS case
Solve ABC with a = 12, b = 27, and c = 20.
SOLUTION
First find the angle opposite the longest side, AC . Use the law of cosines to solve for B.
b2 = a2 + c2 – 2ac cos B
272 = 122 + 202 – 2(12)(20) cos B
272 = 122 + 202
– 2(12)(20)= cos B
– 0.3854 cos B
B cos –1 (– 0.3854) 112.7°
Law of cosines
Substitute.
Solve for cos B.
Simplify.
Use inverse cosine.
![Page 4: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/4.jpg)
EXAMPLE 2 Solve a triangle for the SSS case
Now use the law of sines to find A.
sin Aa =
sin Bb
sin A12
sin 112.7°27
=
sin A =12 sin 112.7°
270.4100
A sin–1 0.4100 24.2°
Law of sines
Substitute for a, b, and B.
Multiply each side by 12 and simplify.
Use inverse sine.
The third angle C of the triangle is C 180° – 24.2° – 112.7° = 43.1°.
In ABC, A 24.2, B 112.7, and C 43.1.ANSWER
![Page 5: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/5.jpg)
EXAMPLE 3 Use the law of cosines in real life
Science
Scientists can use a set of footprints to calculate an organism’s step angle, which is a measure of walking efficiency. The closer the step angle is to 180°, the more efficiently the organism walked.
The diagram at the right shows a set of footprints for a dinosaur. Find the step angle B.
![Page 6: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/6.jpg)
EXAMPLE 3 Use the law of cosines in real life
SOLUTION
b2 = a2 + c2 – 2ac cos B
3162 = 1552 + 1972 – 2(155)(197) cos B
3162 = 1552 + 1972
– 2(155)(197)= cos B
– 0.6062 cos B
B cos –1 (– 0.6062) 127.3° Use inverse cosine.
Simplify.
Solve for cos B.
Substitute.
Law of cosines
The step angle B is about 127.3°.ANSWER
![Page 7: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/7.jpg)
GUIDED PRACTICE for Examples 1, 2, and 3
Find the area of ABC.
1. a = 8, c = 10, B = 48°
SOLUTION
Use the law of cosines to find side length b.
b2 = a2 + c2 – 2ac cos B
b2 = 82 + 102 – 2(8)(10) cos 48°
b2 57
b2 57 7.55
Law of cosines
Substitute for a, c, and B.
Simplify.
Take positive square root.
![Page 8: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/8.jpg)
GUIDED PRACTICE for Examples 1, 2, and 3
Use the law of sines to find the measure of angle A.
sin Aa
sin Bb
=
sin A 8
=sin 48°7.55
sin A =8 sin 48°
7.550.7874
A sin –1 0.7836 51.6°
Law of sines
Substitute for a, b, and B.
Multiply each side by 8 andsimplify.
Use inverse sine.
The third angle C of the triangle is C 180° – 48° – 52.2° = 79.8°.
In ABC, b 7.55, A 52.2°, and C 94.8°.ANSWER
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162 = 142 + 92 – 2(14)(9) cos B
GUIDED PRACTICE for Examples 1, 2, and 3
Find the area of ABC.
2. a = 14, b = 16, c = 9
SOLUTION
First find the angle opposite the longest side, AC . Use the law of cosines to solve for B.
b2 = a2 + c2 – 2ac cos B
162 = 142 + 92
– 2(14)(9)= cos B
Law of cosines
Substitute.
Solve for cos B.
![Page 10: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/10.jpg)
GUIDED PRACTICE for Examples 1, 2, and 3
– 0.0834 cos B
B cos –1 (– 0.0834) 85.7°
Simplify.
Use inverse cosine.
sin Aa
= sin Bb
sin A14
sin 85.2°16
=
sin A =14sin 85.2°
160.8719
Law of sines
Substitute for a, b, and B.
Multiply each side by 14 and simplify.
Use the law of sines to find the measure of angle A.
![Page 11: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/11.jpg)
GUIDED PRACTICE for Examples 1, 2, and 3
The third angle C of the triangle is C 180° – 85.2° – 60.7° = 34.1°.
A sin–1 0.8719 60.7° Use inverse sine.
In ABC, A 60.7°, B 85.2°, and C 34.1°.
ANSWER
![Page 12: EXAMPLE 1 Solve a triangle for the SAS case Solve ABC with a = 11, c = 14, and B = 34°. SOLUTION Use the law of cosines to find side length b. b 2 = a](https://reader036.vdocument.in/reader036/viewer/2022062713/56649cc15503460f94988656/html5/thumbnails/12.jpg)
GUIDED PRACTICE for Examples 1, 2, and 3
SOLUTION
b2 = a2 + c2 – 2ac cos B
3352 = 1932 + 1862 – 2(193)(186) cos B
3352 = 1932 + 1862
– 2(193)(186)= cos B
– 0.5592 cos B
B cos –1 (– 0.5592) 127° Use inverse cosine.
Simplify.
Solve for cos B.
Substitute.
Law of cosines
3. What If? In Example 3, suppose that a = 193 cm, b = 335 cm, and c = 186 cm. Find the step angle θ.
The step angle B is about 124°.ANSWER