If sin(x) = 4/5 and cos(y) = 5/13 with both x and y terminating in quadrant 1 find the exact value of cos(x-y)
I know that the denominator will be 65
Use Pythagoras for each to find
i
sinx = 4/5, then cosx = 3/5
if cosy = 5/13, then siny = 12/13
cos(x-y) = cosxcosy + sinxsiny
= (3/5)(5/13) + (4/5)(12/13)
= (15+48)/65 = 63/65
Well, if you know the denominator will be 65, I guess that means you have a "cos-numatic" sense!
To find the exact value of cos(x - y), we can use the trigonometric identity cos(x - y) = cos(x)cos(y) + sin(x)sin(y).
Given that sin(x) = 4/5 and cos(y) = 5/13, we can substitute these values into the formula.
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
cos(x - y) = (cos(y) * cos(x)) + (sin(y) * sin(x))
cos(x - y) = (5/13 * cos(x)) + (sin(y) * 4/5)
To find cos(x), we can use the Pythagorean Identity sin^2(x) + cos^2(x) = 1. Since sin(x) = 4/5, we can substitute this value into the formula.
(4/5)^2 + cos^2(x) = 1
16/25 + cos^2(x) = 1
cos^2(x) = 1 - 16/25
cos^2(x) = 9/25
Taking the square root of both sides, we get cos(x) = ±(3/5). Since x is in quadrant 1, we take the positive value, so cos(x) = 3/5.
Now substituting this value into the equation for cos(x - y), we get
cos(x - y) = (5/13 * 3/5) + (sin(y) * 4/5)
cos(x - y) = 15/65 + (sin(y) * 4/5)
To find sin(y), we can use the Pythagorean Identity sin^2(y) + cos^2(y) = 1. Since cos(y) = 5/13, we can substitute this value into the formula.
sin^2(y) + (5/13)^2 = 1
sin^2(y) + 25/169 = 1
sin^2(y) = 1 - 25/169
sin^2(y) = 144/169
Taking the square root of both sides, we get sin(y) = ±(12/13). Since y is in quadrant 1, we take the positive value, so sin(y) = 12/13.
Now substituting this value into the equation for cos(x - y), we get
cos(x - y) = 15/65 + (12/13 * 4/5)
cos(x - y) = 15/65 + 48/65
cos(x - y) = 63/65
Therefore, the exact value of cos(x - y) is 63/65.
To find the exact value of cos(x-y), we can use the trigonometric identity cos(x-y) = cos(x)cos(y) + sin(x)sin(y).
Given that sin(x) = 4/5 and cos(y) = 5/13, we can substitute these values into the identity.
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
= (4/5)(5/13) + (cos(y)/sin(y))(4/5)
Next, we need to find cos(y)/sin(y). We can use the Pythagorean identity sin^2(y) + cos^2(y) = 1 to find sin(y).
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 - 25)/169
sin^2(y) = 144/169
sin(y) = sqrt(144/169)
sin(y) = 12/13
Now we can substitute the values of cos(y) = 5/13 and sin(y) = 12/13 into our equation for cos(x-y).
cos(x-y) = (4/5)(5/13) + (cos(y)/sin(y))(4/5)
= 20/65 + (5/13)(4/5)
= 20/65 + 20/65
= 40/65
Simplifying the fraction, we get:
cos(x-y) = 40/65
Therefore, the exact value of cos(x-y) is 40/65.