If cosx=4/5, cosy=12/13 in Quad. IV, find sin(x+y)
To find sin(x+y), we can use the trigonometric identities. One of the fundamental identities that we can use here is the sum of angles formula for sine:
sin(x+y) = sinx*cosy + cosx*siny
We are given that cosx = 4/5 and cosy = 12/13. Since cos(x) is positive and cosine is positive in the fourth quadrant, we can determine that sinx is negative.
Step 1: Determine sinx
We can find sinx using the Pythagorean identity:
sin^2(x) = 1 - cos^2(x)
sin^2(x) = 1 - (4/5)^2
sin^2(x) = 1 - 16/25
sin^2(x) = 9/25
Since sinx is negative in the fourth quadrant, sinx = -(√(9/25)) = -3/5
Step 2: Determine siny
Using the same method as above, we find:
sin^2(y) = 1 - cos^2(y)
sin^2(y) = 1 - (12/13)^2
sin^2(y) = 1 - 144/169
sin^2(y) = 25/169
Since cos y is positive and cosine is positive in the fourth quadrant, siny is positive. Thus, siny = √(25/169) = 5/13
Step 3: Calculate sin(x+y)
Now, we can substitute the values we found into the sum of angles formula:
sin(x+y) = sinx * cosy + cosx * siny
sin(x+y) = (-3/5) * (12/13) + (4/5) * (5/13)
sin(x+y) = (-36/65) + (20/65)
sin(x+y) = -16/65
Therefore, sin(x+y) = -16/65.