If sin(x) = /45 and cos(y) = 5/13 with both x and y terminating in quadrant 1 find the exact value of cos(x-y)
cos(4/5 - 5/13)
Is this what I would do?
To find the exact value of cos(x-y), where sin(x) = 4/5 and cos(y) = 5/13, you can use the trigonometric identity for cos(x-y):
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
Given that sin(x) = 4/5 and cos(y) = 5/13, you can substitute the values into the formula:
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
cos(x-y) = (sqrt(1 - sin^2(x))) * cos(y) + sin(x) * sin(y)
cos(x-y) = (sqrt(1 - (4/5)^2)) * (5/13) + (4/5) * (5/13)
Now, let's simplify the expression:
cos(x-y) = (sqrt(1 - (16/25))) * (5/13) + (20/65)
cos(x-y) = (sqrt(9/25)) * (5/13) + (20/65)
cos(x-y) = (3/5) * (5/13) + (20/65)
cos(x-y) = 15/65 + 20/65
cos(x-y) = 35/65
So, the exact value of cos(x-y) is 35/65, which can be simplified further if necessary.