Question 304701: A boat is traveling in a triangular manner. The boat traveled 213 miles from point a to point b.It then took a 23 degree turn going from point b to point c and traveled 105 miles to get there. At what angle must it turn to get from point c to point a?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Since you are given 2 sides and the included angle, I believe you would use the Law of Cosines to solve this.
If you label your triangle ABC and you make angle B equal to 23 degrees, and you label each side opposite their respective angles a,b,c, then you will have:
Side a is opposite angle A
Side b is opposite angle B
Side c is opposite angle C
Side a is between points B and C.
Side b is between points A and C.
Side c is between points A and B.
This would make side c = 213.
This would make side a = 105.
Since you want to find the length of b and you are given angle B, then the Law of Cosines formula you would use would be:
Substituting known values in this equation gets:
Simplify this equation to get:
 which becomes:
 .
Take the square root of both sides of this equation to get:
b = 123.3686261
You can now use either the Law of Cosines or the Law of Sines to get the remaining angles of this triangle.
Using the Law of Cosines, angle C would be found using the following formula:
Solve this equation for cos(C) to get:
Substitute known values in this equation to get:
This would result in cos(C) = .738174175 which would result in:
angle C = 42.42388532 degrees.
Using the Law of Sines, angle C would be found using the following formula:
Substituting known values in this equation gets:
Cross multiply to get:
Divide both sides of this equation by 123.3686261 to get:
 which becomes:
sin(C) = .674610174 which results in:
angle C = 42.42388532 degrees.
You got the same answer for angle C either way which is a good sign that you did all your calculations correctly.
Using the Law of Cosines, you got angle C = 42.42388532 degrees.
Using the Law of Sines, you got angle C = 42.42388532 degrees.
Since all 3 angles of the triangle must total up to 180 degrees, the remaining angle A is equal to:
180 - 23 - 42.42388532 = 114.5761147 degrees.
Your 3 sides are:
a = 105
b = 123.3686261
c = 213
Your 3 angles are:
A = 114.5761147 degrees
B = 23 degrees
C = 42.42388532 degrees
A picture of your triangle is shown below:
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