if x and y are positive acute angles, tan x=1/3and tan y =1/4, find the value of sin(x-y)
To find the value of sin(x-y), we can use the trigonometric identity: sin(x-y) = sin(x)cos(y) - cos(x)sin(y).
Given that tan x = 1/3 and tan y = 1/4, we can use the relationships involving the trigonometric functions to find the values of sin x, cos x, sin y, and cos y.
1. Recall that tan x = sin x / cos x. From tan x = 1/3, we can solve for sin x and cos x:
- sin x = 1; cos x = 3.
2. Similarly, tan y = sin y / cos y. From tan y = 1/4, we can solve for sin y and cos y:
- sin y = 1; cos y = 4.
Now, we have all the values necessary to calculate sin(x-y):
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)
= (1)(4) - (3)(1)
= 4 - 3
= 1.
Hence, the value of sin(x-y) is 1.