example 1 solve a triangle for the sas case solve abc with a = 11, c = 14, and b = 34°. solution...
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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.
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
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.
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
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.
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
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.
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
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.
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.
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
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