Angle x is in the first quadrant and angle y is in the second quadrant, such that Cosx = 12/13 and Siny = 7/25. Determine an exact value for Cos(x+y).
If you draw the right triangles for x and y in standard position, you can see that
sinx = 5/13
cosy = -24/25
Now just use your sum formula
cos(x+y) = cosx cosy - sinx siny
= (12/13)(-24/25) - (5/13)(7/25) = -323/325
Well, it seems like angle x and angle y have decided to team up for an interesting math puzzle. Let's see if we can solve it!
First, we'll need to find the value of Sinx, since we know Cosx = 12/13. To do that, we can use the identity Sin^2x + Cos^2x = 1. Plugging in the value of Cosx, we have:
Sin^2x + (12/13)^2 = 1
Simplifying, we get:
Sin^2x + 144/169 = 1
Sin^2x = 169/169 - 144/169
Sin^2x = 25/169
Sin x = √(25/169)
Sin x = 5/13
So now we know Sin x = 5/13 and Sin y = 7/25. To find Cos(x+y), we can use the identity Cos(x+y) = Cos x * Cos y - Sin x * Sin y. Plugging in the values, we have:
Cos(x+y) = (12/13)*(√(1-(5/13)^2)) - (5/13)*(7/25)
Simplifying further:
Cos(x+y) = (12/13)*(√(1-25/169)) - (5/13)*(7/25)
Cos(x+y) = (12/13)*(√(169/169-25/169)) - (5/13)*(7/25)
Cos(x+y) = (12/13)*(√(144/169)) - (5/13)*(7/25)
Cos(x+y) = (12/13)*(12/13) - (5/13)*(7/25)
Cos(x+y) = 144/169 - 35/169
Cos(x+y) = 109/169
So the exact value of Cos(x+y) is 109/169. Voilà!
To find the exact value of cos(x+y), we can use the cosine addition formula:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Given that cos(x) = 12/13 and sin(y) = 7/25, we need to find the value of cos(y).
Since angle x is in the first quadrant and cos(x) = 12/13, we can use the Pythagorean identity to find sin(x):
sin(x) = √(1 - cos^2(x))
sin(x) = √(1 - (12/13)^2)
sin(x) = √(1 - 144/169)
sin(x) = √(169/169 - 144/169)
sin(x) = √(25/169)
sin(x) = 5/13
Now, since sin(y) = 7/25 is in the second quadrant, we can use the Pythagorean identity again to find cos(y):
cos(y) = √(1 - sin^2(y))
cos(y) = √(1 - (7/25)^2)
cos(y) = √(1 - 49/625)
cos(y) = √(625/625 - 49/625)
cos(y) = √(576/625)
cos(y) = 24/25
Now, we can substitute the values of cos(x), sin(x), and cos(y) into the cosine addition formula:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x+y) = (12/13)(24/25) - (5/13)(7/25)
cos(x+y) = (12*24)/(13*25) - (5*7)/(13*25)
cos(x+y) = 288/325 - 35/325
cos(x+y) = 253/325
Therefore, the exact value of cos(x+y) is 253/325.
To determine an exact value for Cos(x+y), we can use the trigonometric identity for the cosine of the sum of two angles:
Cos(x+y) = Cosx * Cosy - Sinx * Siny
Given that Cosx = 12/13 and Siny = 7/25, we need to find the values of Cosy and Sinx to substitute into the formula.
Since angle x is in the first quadrant and Cosx is positive, we know that Sinx will also be positive, since Sinx = √(1 - Cosx^2).
Using the Pythagorean identity for Sinx, we can find Sinx:
Sin^2x = 1 - Cos^2x
Sin^2x = 1 - (12/13)^2
Sin^2x = 1 - 144/169
Sin^2x = (169 - 144)/169
Sin^2x = 25/169
Sin x = √(25/169)
Sin x = 5/13 (Since Sinx > 0 in the first quadrant)
Next, we'll find Cosy. Since angle y is in the second quadrant, Cosy will be negative. We can find Cosy using the Pythagorean identity for Cosy:
Cos^2y = 1 - Sin^2y
Cos^2y = 1 - (7/25)^2
Cos^2y = 1 - 49/625
Cos^2y = (625 - 49)/625
Cos^2y = 576/625
Cos y = -√(576/625)
Cos y = -24/25 (Since Cosy < 0 in the second quadrant)
Finally, substitute the values of Cosx, Siny, Cosy, and Sinx into the formula for Cos(x+y):
Cos(x+y) = Cosx * Cosy - Sinx * Siny
Cos(x+y) = (12/13) * (-24/25) - (5/13) * (7/25)
Cos(x+y) = -288/325 - 35/325
Cos(x+y) = (-288 - 35)/325
Cos(x+y) = -323/325
Therefore, the exact value for Cos(x+y) is -323/325.