math

Angles x and y are located in the first quadrant such that sinx= 4/5 and cosy= 7/25.

determine an exact value for xin(x+Y)

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Determine an exact value for sin ( x + y )

then:

sin ( x + y ) = sin x ∙ cos y + cos x ∙ sin y

sin x = 4 / 5

cos x = ± √ ( 1 - sin² x )

In the first quadrant, all trigonometric functions are positive, so:

cos x = √ ( 1 - sin² x )

cos x = √ [ 1 - ( 4 / 5 )² ]

cos x = √ ( 1 - 16 / 25 )

cos x = √ ( 25 / 25 - 16 / 25 )

cos x = √ ( 9 / 25 )

cos x = √ 9 / √ 25

cos x = 3 / 5

cos y = 7 / 25

sin y = ± √ ( 1 - cos² y )

In the first quadrant, all trigonometric functions are positive, so:

sin y = √ ( 1 - cos² y )

sin y = √ [ 1 - ( 7 / 25 )² y )

sin y = √ ( 1 - 49 / 625 )

sin y = √ ( 625 / 625 - 49 / 625 )

sin y = √ ( 576 / 625 )

sin y = √ 576 / √ 625

sin y = 24 / 25

sin ( x + y ) = sin x ∙ cos y + cos x ∙ sin y

sin ( x + y ) = 4 / 5 ∙ 7 / 25 + 3 / 5 ∙ 24 / 25

sin ( x + y ) = 28 / 125 + 72 / 125

sin ( x + y ) = 100 / 125

sin ( x + y ) = 25 ∙ 4 / 25 ∙ 5

sin ( x + y ) = 4 / 5

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