if tan x= 4/3 and pip < x < 3pi/2, and cot y= -5/12 with 3pi/2 < y < 2pi find sin(x-y)

To find sin(x-y), we need to know the values of sin(x) and sin(y). We can calculate them using the given information.

Given:
tan x = 4/3 (π < x < (3π/2))
cot y = -5/12 (3π/2 < y < 2π)

To find sin(x), we can use the fact that tan x = sin x / cos x:
tan x = sin x / cos x
4/3 = sin x / cos x

To find sin y, we can use the fact that cot y = cos y / sin y:
cot y = cos y / sin y
-5/12 = cos y / sin y

Since both equations involve sin and cos, we can solve them simultaneously:

From equation 1: sin x = (4/3) cos x
From equation 2: cos y = (-5/12) sin y

Now we can substitute these equations into sin(x-y):

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

Substituting sin x = (4/3) cos x and cos y = (-5/12) sin y:

sin(x-y) = (4/3)cos(x)(-5/12)sin(y) - cos(x)(-5/12)sin(y)
= (-20/36) sin(x) sin(y) + (5/12) cos(x) sin(y)

Now, we need to substitute the values of sin x and sin y:

sin(x-y) = (-20/36) [(4/3) cos(x)] [(-5/12) sin(y)] + (5/12) cos(x) [(-5/12) sin(y)]

Simplifying further:

sin(x-y) = (-20/36) (4/3) (-5/12) sin(x) sin(y) + (5/12) (5/12) cos(x) sin(y)
= (100/648) sin(x) sin(y) + (25/144) cos(x) sin(y)

Finally, substitute the values sin x = (4/3) cos x and sin y = (-5/12) cos y:

sin(x-y) = (100/648) [(4/3) cos x] [(-5/12) cos y] + (25/144) cos(x) [(-5/12) cos y]

Simplifying further:

sin(x-y) = (20/81) cos(x) cos(y) + (25/144) cos(x) cos(y)

Therefore, sin(x-y) = (20/81) cos(x) cos(y) + (25/144) cos(x) cos(y).

To find sin(x-y), we first need to find the values of x and y. We are given that tan(x) = 4/3 and cot(y) = -5/12, along with the given ranges for x and y.

Let's start with finding x.

Given that tan(x) = 4/3, we can use the inverse tangent function (arctan) to find x.

arctan(tan(x)) = arctan(4/3)

Now, let's find the principal value of x within the given range.

Since pi < x < 3pi/2, x lies in the third quadrant. In the third quadrant, tan(x) is negative. Therefore, we can write:

x = arctan(-4/3) + pi

Similarly, let's find y.

Given that cot(y) = -5/12, we can use the inverse cotangent function (arccot) to find y.

arccot(cot(y)) = arccot(-5/12)

Now, let's find the principal value of y within the given range.

Since 3pi/2 < y < 2pi, y lies in the fourth quadrant. In the fourth quadrant, cot(y) is positive. Therefore, we can write:

y = arccot(5/12) + 2pi

Now that we have found the values of x and y, we can calculate sin(x-y) using the trigonometric identity:

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

To evaluate sin(x) and cos(x), we need to determine the signs based on the quadrant in which x lies.

Since x lies in the third quadrant (pi < x < 3pi/2), sin(x) will be negative.

sin(x) = -√(1 - cos^2(x))

Similarly, cos(x) will be negative.

cos(x) = -√(1 - sin^2(x))

To evaluate sin(y) and cos(y), we need to determine the signs based on the quadrant in which y lies.

Since y lies in the fourth quadrant (3pi/2 < y < 2pi), sin(y) will be negative.

sin(y) = -√(1 - cos^2(y))

Similarly, cos(y) will be positive.

cos(y) = √(1 - sin^2(y))

Now, substitute these values into the formula sin(x-y) = sin(x)cos(y) - cos(x)sin(y) and calculate the result.