In triangle(ABC), sin(A) : sin(B) : sin(C) = 2 : 3 : 4 Find cos(A + C)
In triangle(ABC), let side a = 2x side b = 3x side c = 4x
Using the Law of Cosines: cos(C) = ( a2 + b2 - c2 ]) / ( 2·a·b )
cos(A) = [ (3x)2 + (4x)2 - (2x)2 ] / [ 2·(3x)·(4x) ] = [ 9x2 + 16x2 - 4x2 ] / [ 24x2 ] = 7/8
Using the Pythagorean Theorem: sin(A) = sqrt(15) / 8
cos(C) = [ (2x)2 + (3x)2 - (4x)2 ] / [ 2·(2x)·(3x) ] = [ -3x2 ] / [ 12x2 ] = -1/4
Using the Pythagorean Theorem: sinC) = sqrt(15) / 4
Using the formula: cos(A + C) = cos(A) ·cos(C) - sin(A)·sin(C)
= (7/8) · (-1/4) - ( sqrt(15) / 8 ) · ( sqrt(15) / 4 )
= -11/16
In triangle(ABC), sin(A) : sin(B) : sin(C) = 2 : 3 : 4 Find cos(A + C)
In triangle(ABC), let side a = 2x side b = 3x side c = 4x
Using the Law of Cosines: cos(C) = ( a2 + b2 - c2 ]) / ( 2·a·b )
cos(A) = [ (3x)2 + (4x)2 - (2x)2 ] / [ 2·(3x)·(4x) ] = [ 9x2 + 16x2 - 4x2 ] / [ 24x2 ] = 7/8
Using the Pythagorean Theorem: sin(A) = sqrt(15) / 8
cos(C) = [ (2x)2 + (3x)2 - (4x)2 ] / [ 2·(2x)·(3x) ] = [ -3x2 ] / [ 12x2 ] = -1/4
Using the Pythagorean Theorem: sinC) = sqrt(15) / 4
Using the formula: cos(A + C) = cos(A) ·cos(C) - sin(A)·sin(C)
= (7/8) · (-1/4) - ( sqrt(15) / 8 ) · ( sqrt(15) / 4 )
= -11/16