If angle x and y are in the same quadrant , and sin x=3/5 and cos y=5/13, determine the value of sin(x-y) .

well, you know that

sinx = 3/5
cosx = 4/5

siny = 12/13
cosy = 5/13

Now just use the identity

sin(x-y) = sinx cosy - cosx siny

To determine the value of sin(x-y), we need to use the trigonometric identity for the difference of angles:

sin(x - y) = sin(x) * cos(y) - cos(x) * sin(y)

Given that sin(x) = 3/5 and cos(y) = 5/13, we need to find the values of cos(x) and sin(y) in order to evaluate sin(x - y).

To find cos(x), we can use the Pythagorean identity:

cos^2(x) + sin^2(x) = 1

Substituting sin(x) = 3/5 into the identity and solving for cos(x):

cos^2(x) + (3/5)^2 = 1
cos^2(x) + 9/25 = 1
cos^2(x) = 1 - 9/25
cos^2(x) = 25/25 - 9/25
cos^2(x) = 16/25
cos(x) = ±√(16/25)
cos(x) = ±(4/5)

Since x and y are in the same quadrant, both x and y must be positive angles. Therefore, cos(x) cannot be negative. Hence, we take the positive value:

cos(x) = 4/5

To find sin(y), we can use the Pythagorean identity:

cos^2(y) + sin^2(y) = 1

Substituting cos(y) = 5/13 into the identity and solving for sin(y):

(5/13)^2 + sin^2(y) = 1
25/169 + sin^2(y) = 1
sin^2(y) = 1 - 25/169
sin^2(y) = 169/169 - 25/169
sin^2(y) = 144/169
sin(y) = ±√(144/169)
sin(y) = ±(12/13)

Since x and y are in the same quadrant, both x and y must be positive angles. Therefore, sin(y) cannot be negative. Hence, we take the positive value:

sin(y) = 12/13

Now we can substitute the values into the formula for sin(x - y):

sin(x - y) = sin(x) * cos(y) - cos(x) * sin(y)
sin(x - y) = (3/5) * (5/13) - (4/5) * (12/13)
sin(x - y) = (3/5) * (5/13) - (4/5) * (12/13)
sin(x - y) = 15/65 - 48/65
sin(x - y) = -33/65

Therefore, the value of sin(x - y) is -33/65.