Angles x and y are located in the first quadrant such that sinx=3/5 and cosy=5/13.

a) Determine an exact value for cosx
b) Determine an exact value for siny

a)

let x = the x length
let y = the y length
let r = the length from the origin to the x,y point.

r^2 = x2 + y^2

sine of x = y/r
so y and r are given in the problem.
solve for x.
cosine of x = x/r

In the explanation, the angle x and the x length are different items.

To find the exact value of cosx, we need to use the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. Given that sinx = 3/5, we can solve for cosx as follows:

sin^2(x) + cos^2(x) = 1
(3/5)^2 + cos^2(x) = 1
9/25 + cos^2(x) = 1
cos^2(x) = 1 - 9/25
cos^2(x) = 25/25 - 9/25
cos^2(x) = 16/25

Since cosine is positive in the first quadrant, we take the positive square root of 16/25.

cosx = sqrt(16/25)
cosx = 4/5

Therefore, the exact value of cosx is 4/5.

To find the exact value of siny, we can use the same approach. Given that cos(y) = 5/13, we can solve for siny as follows:

sin^2(y) + cos^2(y) = 1
sin^2(y) + (5/13)^2 = 1
sin^2(y) + 25/169 = 1
sin^2(y) = 1 - 25/169
sin^2(y) = 169/169 - 25/169
sin^2(y) = 144/169

Since sine is positive in the first quadrant, we take the positive square root of 144/169.

siny = sqrt(144/169)
siny = 12/13

Therefore, the exact value of siny is 12/13.