Given sin a = 15/17 with a in Quadrant I and cos beta = 4/5 with beta in QuandrantIV, find the exact value of cos(a+beta)
sin a = 15/17 , with a in I, so cos a = 8/17
cos b = 4/5, with b in IV, then sin b = -3/5
cos(a+b) = cosa cosb - sina sinb
= (8/17)(4/5) - (15/17)(-3/5)
= (32 + 45)/85
= 77/85
Well, cos(a + beta) can be found using the cosine addition formula: cos(a + beta) = cos(a) * cos(beta) - sin(a) * sin(beta).
So, let's plug in the given values. We know that sin(a) = 15/17 and cos(beta) = 4/5.
Now, we just need to find cos(a). Since sin(a) = 15/17, we can use the Pythagorean identity sin^2(a) + cos^2(a) = 1 to find cos(a).
sin^2(a) + cos^2(a) = 1
(15/17)^2 + cos^2(a) = 1
225/289 + cos^2(a) = 1
cos^2(a) = 289/289 - 225/289
cos^2(a) = 64/289
Taking the square root of both sides, we get:
cos(a) = ±8/17
Since a is in Quadrant I, cos(a) is positive. So cos(a) = 8/17.
Now, let's plug in all the values into the cosine addition formula:
cos(a + beta) = cos(a) * cos(beta) - sin(a) * sin(beta)
cos(a + beta) = (8/17) * (4/5) - (15/17) * (3/5)
Now, let the clowns do the math:
cos(a + beta) = (8/17) * (4/5) - (15/17) * (3/5)
= 32/85 - 45/85
= -13/85
So, the exact value of cos(a + beta) is -13/85.
To find the exact value of cos(a + beta), we can use the cosine addition formula:
cos(a + beta) = cos a * cos beta - sin a * sin beta
Given sin a = 15/17 and cos beta = 4/5:
sin a = 15/17
cos beta = 4/5
To find cos a, we can use the Pythagorean Identity:
cos^2 a + sin^2 a = 1
(1 - sin^2 a) + sin^2 a = 1
1 - sin^2 a + sin^2 a = 1
1 = 1
Since sin a = 15/17, we can find cos a as:
cos a = √(1 - sin^2 a)
cos a = √(1 - (15/17)^2)
cos a = √(1 - 225/289)
cos a = √(289/289 - 225/289)
cos a = √(64/289)
cos a = 8/17
Now we can substitute the given values into the cosine addition formula:
cos(a + beta) = cos a * cos beta - sin a * sin beta
cos(a + beta) = (8/17) * (4/5) - (15/17) * (0)
cos(a + beta) = (32/85) - 0
cos(a + beta) = 32/85
Therefore, the exact value of cos(a + beta) is 32/85.
To find the exact value of cos(a + beta), we can use the cosine addition formula:
cos(A + B) = cos A * cos B - sin A * sin B
First, let's find the values of sin a and cos beta:
sin a = 15/17 (given in the question)
cos beta = 4/5 (given in the question)
To find cos a, we can use the Pythagorean identity:
cos^2 a + sin^2 a = 1
Since a is in Quadrant I, sin a is positive. Therefore:
sin^2 a = (15/17)^2 = 225/289
cos^2 a = 1 - sin^2 a = 1 - 225/289 = 64/289
cos a = +/- sqrt(64/289)
Since a is in Quadrant I, cos a is positive. Hence:
cos a = sqrt(64/289) = 8/17
Now, let's use the cosine addition formula:
cos(a + beta) = cos a * cos beta - sin a * sin beta
Substituting the given values:
cos(a + beta) = (8/17) * (4/5) - (15/17) * sin beta
We still need the value of sin beta. To find sin beta, we can use the Pythagorean identity once again:
sin^2 beta + cos^2 beta = 1
Since beta is in Quadrant IV, cos beta is positive. Therefore:
cos^2 beta = (4/5)^2 = 16/25
sin^2 beta = 1 - cos^2 beta = 1 - 16/25 = 9/25
sin beta = -sqrt(9/25) = -3/5
Now, substituting the values:
cos(a + beta) = (8/17) * (4/5) - (15/17) * (-3/5)
Simplifying:
cos(a + beta) = (32/85) + (45/85)
cos(a + beta) = 77/85
Therefore, the exact value of cos(a + beta) is 77/85.