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.
To determine the values of cos(x) and sin(y), we can use the Pythagorean identity and the given information.
a) To find cos(x), we can use the Pythagorean identity:
cos^2(x) = 1 - sin^2(x)
Given that sin(x) = 3/5, we can substitute this into the equation:
cos^2(x) = 1 - (3/5)^2
cos^2(x) = 1 - 9/25
cos^2(x) = 25/25 - 9/25
cos^2(x) = 16/25
Taking the square root of both sides, we have:
cos(x) = ±sqrt(16/25)
Since angle x is in the first quadrant, where cosine is positive, we can take the positive value:
cos(x) = sqrt(16/25) = 4/5
b) To find sin(y), we can use the Pythagorean identity:
sin^2(y) = 1 - cos^2(y)
Given that cos(y) = 5/13, we can substitute this into the equation:
sin^2(y) = 1 - (5/13)^2
sin^2(y) = 1 - 25/169
sin^2(y) = 169/169 - 25/169
sin^2(y) = 144/169
Taking the square root of both sides, we have:
sin(y) = ±sqrt(144/169)
Since angle y is in the first quadrant, where sine is positive, we can take the positive value:
sin(y) = sqrt(144/169) = 12/13
Therefore, the exact values for cos(x) and sin(y) are 4/5 and 12/13, respectively.
To find the exact value of cosx, we can use the Pythagorean Identity for sinx and cosx:
sin^2(x) + cos^2(x) = 1
Since we know sinx = 3/5, we can substitute this value into the equation:
(3/5)^2 + cos^2(x) = 1
9/25 + cos^2(x) = 1
cos^2(x) = 1 - 9/25
cos^2(x) = 16/25
Now, take the square root of both sides to solve for the exact value of cosx:
cosx = ± √(16/25)
Since x is in the first quadrant, cosx should be positive. Therefore:
cosx = √(16/25)
cosx = 4/5 (Exact value of cosx)
To find the exact value of siny, we can use the same process. Since cosy = 5/13, we know:
sin^2(y) + cos^2(y) = 1
Plugging in the value for cosy:
sin^2(y) + (5/13)^2 = 1
sin^2(y) = 1 - 25/169
sin^2(y) = 144/169
Taking the square root of both sides:
siny = ± √(144/169)
Since y is in the first quadrant, siny should be positive. Therefore:
siny = √(144/169)
siny = 12/13 (Exact value of siny)
x is in a 3-4-5 triangle
y is in a 12-13-5 triangle