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.