Question 304701
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:


{{{b^2 = a^2 + c^2 - 2*a*c*cos(B)}}}


Substituting known values in this equation gets:


{{{b^2 = (105)^2 + (213)^2 - 2*105*213*cos(B)}}}


Simplify this equation to get:


{{{b^2 = 11025 + 45363 - 41174.18209}}} which becomes:


{{{b^2 = 15219.81791}}}.


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:


{{{c^2 = a^2 + b^2 - 2*a*b*cos(C)}}}


Solve this equation for cos(C) to get:


{{{cos(C) = (c^2 - a^2 - b^2)/(2*a*b)}}}


Substitute known values in this equation to get:


{{{cos(C) = ((213)^2 - (105)^2 - (123.3686261)^2)/(2*105*123.36862461)}}}


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:


{{{b/sin(B) = c/sin(C)}}}


Substituting known values in this equation gets:


{{{123.3686261/sin(23) = 213/sin(C)}}}


Cross multiply to get:


{{{123.3686261*sin(C) = 213*sin(23)}}}


Divide both sides of this equation by 123.3686261 to get:


{{{sin(C) = (213*sin(23))/123.3686261}}} 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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