If A and B are acute angle such that SinA=8/17 and CosB=3/5.Find
1, Cos(A+B)
2, Sin(A+B)
3, Sin(A-B)
Is this why it stop
Correct
To find the values of trigonometric functions, we can use basic trigonometric identities and properties. Let's solve each part of the question step by step:
1. Finding cos(A+B):
To find cos(A+B), we can use the cosine addition formula:
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
Given that sin(A) = 8/17 and cos(B) = 3/5, we need to find cos(A) and sin(B) first.
To find sin(B), we can use the Pythagorean identity sin^2(B) + cos^2(B) = 1. Since we already know cos(B) = 3/5, we can solve for sin(B):
sin^2(B) + (3/5)^2 = 1
sin^2(B) + 9/25 = 1
sin^2(B) = 1 - 9/25
sin^2(B) = 16/25
sin(B) = sqrt(16/25) = 4/5
Similarly, we can find cos(A) using the Pythagorean identity sin^2(A) + cos^2(A) = 1. Since we already know sin(A) = 8/17, we can solve for cos(A):
(8/17)^2 + cos^2(A) = 1
64/289 + cos^2(A) = 1
cos^2(A) = 1 - 64/289
cos^2(A) = 225/289
cos(A) = sqrt(225/289) = 15/17
Now, we have sin(B) = 4/5, cos(A) = 15/17, cos(B) = 3/5, and sin(A) = 8/17.
Substituting these values into the cosine addition formula:
cos(A+B) = (15/17)(3/5) - (8/17)(4/5)
cos(A+B) = 45/85 - 32/85
cos(A+B) = 13/85
Therefore, cos(A+B) = 13/85.
2. Finding sin(A+B):
To find sin(A+B), we can use the sine addition formula:
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
Using the values we already found, sin(A) = 8/17, cos(B) = 3/5, sin(B) = 4/5, and cos(A) = 15/17:
sin(A+B) = (8/17)(3/5) + (15/17)(4/5)
sin(A+B) = 24/85 + 60/85
sin(A+B) = 84/85
Therefore, sin(A+B) = 84/85.
3. Finding sin(A-B):
To find sin(A-B), we can use the sine subtraction formula:
sin(A-B) = sin(A)cos(B) - cos(A)sin(B)
Using the values we already found, sin(A) = 8/17, cos(B) = 3/5, sin(B) = 4/5, and cos(A) = 15/17:
sin(A-B) = (8/17)(3/5) - (15/17)(4/5)
sin(A-B) = 24/85 - 60/85
sin(A-B) = -36/85
Therefore, sin(A-B) = -36/85.
sinA = 8/17 = Y/r.
X^2 + Y^2 = r^2
X^2 + 8^2 = (17)^2
X^2 = (17)^2 - 8^2 = 225
X = 15.
cosA = X/r = 15/17.
cosB = 3/5 = X/r.
X^2 + Y^2 = r^2.
3^2 + Y^2 = 5^2
Y^2 = 5^2 - 3^2 = 16
Y = 4.
sinB = Y/r = 4/5.
1. cos(A+B) = cosA*cosB - sinA*sinB.
cos(A+B) = 15/17 * 3/5 - 8/17 * 4/5 =
45/85 - 32/85 = 13/85.
2. sin(A+B) = sinA*cosB + cosA*sinB.
The student can solve #2, and #3.