Knowing that 0 <= x <= pi/2, 0 <= y <= pi/2, that sinx = 1/4 and that cosy = 1/5, find sin(x + y), sin(x - y), cos(x + y), cos(x - y)
x and y in first quadrant
if sin x = 1/4, we will need cos x
sin^2 x + cos^2 x = 1
cos x = sqrt( 1-1/16) = sqrt (15/16) = (1/4) sqrt 15
similarly if cos y = 1/5
sin y = sqrt (1- 1/25) =sqrt (24/25)
= (1/5)sqrt 24
now with
sin x = .25, cos x = .25 sqrt 15
and
sin y = .20 sqrt 24, cos y =1/5
Use your trig identities to get the things you want. Watch what quadrant x+y and x-y are in
actually, given the proper values for sin/cos of x and y, the functions of x+y and x-y will work themselves out correctly.
To find the values of sin(x + y), sin(x - y), cos(x + y), and cos(x - y), we need to use trigonometric identities.
First, let's find cosx and siny using the given information.
Given: sinx = 1/4 and cosy = 1/5
We know that sin^2(x) + cos^2(x) = 1 (Pythagorean identity). We can substitute the value of sinx and solve for cosx.
sin^2(x) + cos^2(x) = 1
(1/4)^2 + cos^2(x) = 1
1/16 + cos^2(x) = 1
cos^2(x) = 1 - 1/16
cos^2(x) = 15/16
Taking the square root of both sides, we get:
cos(x) = ±√(15/16)
Since 0 <= x <= pi/2, the value of cos(x) must be positive, so:
cos(x) = √(15/16)
cos(x) = √15/4
Similarly, using the given information cosy = 1/5, we can find siny.
sin^2(y) + cos^2(y) = 1
sin^2(y) + (1/5)^2 = 1
sin^2(y) + 1/25 = 1
sin^2(y) = 1 - 1/25
sin^2(y) = 24/25
Taking the square root of both sides, we get:
sin(y) = ±√(24/25)
Since 0 <= y <= pi/2, the value of sin(y) must be positive, so:
sin(y) = √(24/25)
sin(y) = √24/5
Now, we are ready to calculate sin(x + y), sin(x - y), cos(x + y), and cos(x - y).
1. sin(x + y):
Using the sum-of-angles formula, sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Substituting the known values:
sin(x + y) = (1/4)(1/5) + (√15/4)(√24/5)
sin(x + y) = 1/20 + 2√15/20
sin(x + y) = (1 + 2√15)/20
2. sin(x - y):
Using the difference-of-angles formula, sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
Substituting the known values:
sin(x - y) = (1/4)(1/5) - (√15/4)(√24/5)
sin(x - y) = 1/20 - 2√15/20
sin(x - y) = (1 - 2√15)/20
3. cos(x + y):
Using the sum-of-angles formula, cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
Substituting the known values:
cos(x + y) = (√15/4)(1/5) - (1/4)(√24/5)
cos(x + y) = √15/20 - √6/20
cos(x + y) = (√15 - √6)/20
4. cos(x - y):
Using the difference-of-angles formula, cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
Substituting the known values:
cos(x - y) = (√15/4)(1/5) + (1/4)(√24/5)
cos(x - y) = √15/20 + √6/20
cos(x - y) = (√15 + √6)/20
Therefore, sin(x + y) = (1 + 2√15)/20, sin(x - y) = (1 - 2√15)/20, cos(x + y) = (√15 - √6)/20, and cos(x - y) = (√15 + √6)/20.