Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 10
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43C
cannot be an angle of a triangle with the given sides and angle. This triangle has
2
only one solution. It comes from C
= 25.7°.
1
When C = C
= 25.7°, B = 180° - (70° + 25.7°) = 84.3°. By the Law of Sines, then,
1
°
b
13
13
sin
84
3 .
=
=
=
so
b
13
.
77
.
°
°
°
sin
84
3 .
sin
70
sin
70
The only triangle with sides and angle as given has
a = 13,
b = 13.77, c = 6,
A = 70°, B = 84.3°, C = 25.7°.
Problems:
22. Solve ∆ABC where a = 221, c = 543 and A = 23°.
23. Solve ∆ABC where a = 89.1, c = 100.0 and A = 63°.
24. Solve ∆ABC where a = 9.2, c = 7.6 and A = 98.6°.
25. In oblique triangle ABC, a = 0.7, c = 2.4 and C = 98°. Which one of the following might be angle
A to the nearest tenth of a degree?
A) 3.4°
B) 28.8°
C) 73.2°
D) 163.2°
E) none of these
LAW OF COSINES
For any triangle ABC labeled in the customary
way (as in Figure T22),
2
2
2
a
= b
+ c
- 2bc cos A.
Figure T22
Notice that the Law of Cosines can also be stated as
2
2
2
b
= a
+ c
- 2ac cos B.
and
2
2
2
c
= a
+ b
- 2ab cos C.
When one angle of a triangle is 90°, the Law of Cosines reduces to the Pythagorean Theorem.
Example: Case III: Given two sides and the included angle. Solve ∆ABC where a = 7, b = 9
and C = 47°.
Solution: By the Law of Cosines
2
2
2
c
= 7
+ 9
- 2 · 7 · 9 cos 47°.
Thus
2
c
= 49 + 81 - 85.9318 = 44.0682.
Hence
c = 6.64.
- 10 -
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