Question 1210261
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sin^2(A) + cos^2(A) = 1 ......... pythagorean trig identity
(2/5)^2 + cos^2(A) = 1
4/25 + cos^2(A) = 1
cos^2(A) = 1 - 4/25
cos^2(A) = 21/25
cos(A) = sqrt(21/25) 
cos(A) = sqrt(21)/sqrt(25) 
cos(A) = sqrt(21)/5


sin^2(B) + cos^2(B) = 1
(2/sqrt(7))^2 + cos^2(B) = 1
4/7 + cos^2(B) = 1
cos^2(B) = 1 - 4/7
cos^2(B) = 3/7
cos(B) = sqrt(3/7) 


To summarize so far:
sin(A) = 2/5
cos(A) = sqrt(21)/5
sin(B) = 2/sqrt(7)
cos(B) = sqrt(3/7)


Then,
A+B+C = 180 degrees
C = 180-(A+B)
sin(C) = sin(180 - (A+B) )
sin(C) = sin(A+B)  ........................................... use identity sin(180-x) = sin(x)
sin(C) = sin(A)cos(B) + cos(A)sin(B) ......................... angle sum identity
sin(C) = (2/5)*sqrt(3/7) + (sqrt(21)/5)*(2/sqrt(7))
sin(C) = 2*sqrt(3)/(5*sqrt(7)) + 2*sqrt(21)/(5*sqrt(7))
sin(C) = (2*sqrt(3) + 2*sqrt(21))/(5*sqrt(7))
<font color=red>sin(C) = 2*(sqrt(3) + sqrt(21))/(5*sqrt(7)) exactly
sin(C) = 0.9546817913107 approximately</font>


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Another approach:
sin(A) = 2/5
A = arcsin(2/5)
A = 23.5781784782 degrees approximately
sin(B) = 2/sqrt(7)
B = arcsin(2/sqrt(7))
B = 49.10660535087 degrees approximately


A+B+C = 180
C = 180-A-B
C = 180-23.5781784782-49.10660535087
C = 107.31521617093
sin(C) = sin(107.31521617093)
<font color=red>sin(C) = 0.9546817913107 approximately</font>
The drawback of this method is that you won't be able to determine the exact form in terms of square roots, as shown in the previous section. 
However, some teachers are perfectly fine with the answer in decimal form. 
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