Question 288597
Let your angles be A,B,C.


Let the sides opposite each of those angles be a,b,c.


Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.


angle A is equal to 45 degrees.
angle B is equal to 75 degrees.
angle C is equal to 60 degrees.


You are given that the side opposite the 45 degree angle is 5 units long.


Side a is 5 units long.


The law of Sines states that:


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


This means that the length of each side of a triangle is proportional to the measure of its corresponding angle.


In your triangle, this equation becomes:


{{{5/sin(45) = b/sin(75) = c/sin(60)}}


Use your calculator to find the sine of these angles and then substitute in the equation to get:


{{{5/.707106781 = b/.965925826 = c/.866025404}}}


{{{5/.707106781}}} = {{{7.071067812}}}


The ratio of {{{b/.965925826}}} should be equal to {{{7.071067812}}} which means that {{{b}}} = {{{.965925826 * 7.071067812}}} which equals {{{6.830127019}}}.


The ratio of {{{c/.866025404}}} should be equal to {{{7.071067812}}} which means that {{{c}}} = {{{.866025404 * 7.071067812}}} which equals {{{6.123724357}}}.


The sides of your triangle should be:


a = 5
b = 6.830127019
c = 6.123724357


The law of sines ratio should hold.


That ratio is, once again:


{{{a/sin(A) = b/sin(B) = c/sin(c)}}}.


In your equation, that becomes:


{{{5/sin(45) = 6.830127019/sin(75) = 6.123724357/sin(60)}}}.


Using the sines of those angles previously calculated gets us


{{{7.071067812 = 7.071067812 = 7.071067812}}}


The law of sines has held and your answer is:


a = 5
b = 6.830127019
c = 6.123724357