Answer:
- a = 5.2 units
- A = 53°, B = 44°, C = 83°
- b = 228 feet
Step-by-step explanation:
You have triangles with two sides and the included angle given, or all three sides given. You want the measures of the missing sides or angles.
Law of Cosines
For triangle ABC, the law of cosines tells us ...
c² = a² +b² -2ab·cos(C)
Solving for the third side, we get ...
c = √(a² +b² -2ab·cos(C))
Solving for the angle, we get ...
C = arccos((a² +b² -c²)/(2ab))
The names of the sides and vertices can be permuted, as long as the angle is between the given sides.
1. A=35°, b=7, c=9
a = √((7² +9² -2·7·9·cos(35°)) ≈ 5.176
The measure of side 'a' is about 5.2 units.
2. a=8, b=7, c=10
C = arccos((8² +7² -10²)/(2·8·7)) = arccos(13/112) ≈ 83.33°
A = arcsin((8/10)·sin(83.33°)) ≈ 52.62°
B = arcsin((7/10)·sin(83.33°)) ≈ 44.05°
The last two angles are found using the law of sines:
sin(A)/a = sin(B)/b = sin(C)/c
3. a=230 ft, B=38°, c=360 ft
b = √(230² +360² -2·230·360·cos(38°)) ≈ 228 . . . . feet
The third side measures 228 feet.
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Additional comment
A calculator or spreadsheet can be programmed with the necessary formulas so that you can avoid a lot of repetitive work when you have several of these to solve. You need to make sure angles are treated appropriately, as the default is often radians, rather than degrees.
When solving for angles from three given sides, we like to solve for the largest angle first. That avoids any ambiguity when using the Law of Sines to find the other angles.
When using the Law of Sines to find a second angle after using the Law of Cosines, it can work well to find the smallest remaining angle. (In the first problem, that would be the angle opposite the side of length 7; in the last problem, that would be the angle opposite the side of length 230.)
